Osaka City University Advanced Mathematical Institute (OCAMI)MEXT Joint Usage/Research Center on Mathematics and Theoretical Physics

OCAMI Reports Vol. 2 (2021)
doi: 10.24544/ocu.20210605-002

Submanifolds of Symmetric Spaces
and Their Time Evolutions
Organized by
Kurando Baba
Makoto Kimura
Yoshihiro Ohnita
Homare Tadano
Makiko Sumi Tanaka
Hikaru Yamamoto
Naoyuki Koike (Proposer & Principal Investigator)
March 5 - 6, 2021
Abstract. This workshop held March 5-6, 2021 to conduct international researchexchanges on “Submanifolds of symmetric spaces and their time evolutions”.

2020 Mathematics Subject Classification.
53C35;53C40;53C42;53C44

Key words and Phrases.
submanifold, symmetric space, mean curvature flow

© 2021 OCAMI.
OCAMI. Submanifolds of Symmetric Spaces and Their Time Evolutions. OCAMI Reports. Vol. 2,
Osaka City University Advanced Mathematical Institute. 2021, 173pp.
doi: 10.24544/ocu.20210605-002

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OCAMI Reports Vol. 2 (2021)

PrefaceThis volume of OCAMI Reports summarizes the workshop “Submanifolds of Symmetric Spaces and Their Time Evolutions” held from March 5th to March 6th in 2021
online by Zoom because of the COVID-19 pandemic. This workshop was supported by“Osaka city University, Advanced Mathematical Institute MEXT Joint Usage/ResearchCenter on Mathematics and Theoretical Physics.” The main focus of this workshop is
submanifolds of symmetric spaces (or more general ambient spaces), mean curvature
flows in symmetric spaces (or more general ambient spaces), and related geometricflows (line bundle mean curvature flows , which is flows of connections of complex line
bundle over Kähler manifold, coupling flows of Ricci flows and heat flows, and so on).
This workshop consisted of two 60 minutes keynote lectures on “mean curvature flow
for isoparametric submanifolds and polar foliations on symmetric spaces” by Professor
Xiaobo Liu (Peking University), two 60 minutes keynote lectures on “deformed Hermitian Yang-Mills connections and line bundle mean curvature flows” by Doctor Hikaru
Yamamoto (University of Tsukuba), and seven 50 minutes lectures on “submanifolds ofsymmetric spaces, submanifolds of generalized s-manifolds, proper Fredholm submanifolds of Hilbert spaces, Lagrangian mean curvature flows, and coupling flows of Ricci
flows and heat flows.” There were 38 participants in this workshop. This workshop
conducted international research exchanges on submanifolds of symmetric spaces, theirtime evolutions and furthermore related geometric flows.
March 2021
Naoyuki Koike

Submanifolds of Symmetric Spaces and Their Time Evolutions

3

Organizers
Kurando Baba
Department of Mathematics, Faculty of Science and Technology, Tokyo University of Science,
Noda, Chiba, 278-8510, Japan
E-mail address: baba kurando@ma.noda.tus.ac.jp
Makoto Kimura
Department of Mathematics, Faculty of Science, Ibaraki University, Mito, Ibaraki 310-8512,
Japan
E-mail address: makoto.kimura.geometry@vc.ibaraki.ac.jp
Yoshihiro Ohnita
Osaka City University Advanced Mathematical Institute, & Department of Mathematics,
Osaka City University, 3-3-138 Sugimoto, Sumiyoshi-ku, Osaka, 558-8585, Japan
E-mail address: ohnita@sci.osaka-cu.ac.jp
Homare Tadano
Department of Mathematical Science, Faculty of Science, Yamaguchi University, Yoshida
1677-1, 753-8512 Yamaguchi, Japan
E-mail address: tadano@yamaguchi-u.ac.jp
Makiko Sumi Tanaka
Department of Mathematics, Faculty of Science and Technology, Tokyo University of Science,
Noda, Chiba, 278-8510, Japan
E-mail address: tanaka makiko@ma.noda.tus.ac.jp
Hikaru Yamamoto
Department of Mathematics, Faculty of Pure and Applied Science, University of Tsukuba,
1-1-1 Tennodai, Tsukuba, Ibaraki 305-8577, Japan
E-mail address: hyamamoto@math.tsukuba.ac.jp
Naoyuki Koike (Proposer & Principal Investigator)
Department of Mathematics, Faculty of Science, Tokyo University of Science, 1-3 Kagurazaka
Shinjuku-ku, Tokyo 162-8601 Japan
E-mail address: koike@rs.tus.ac.jp

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OCAMI Reports Vol. 2 (2021)

Table of Contents
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Organizers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2
3

Xiaobo Liu
Ancient solutions for mean curvature flow of isoparametric submanifolds . .

5

Xiaobo Liu
Polar foliations in symmetric spaces and their mean curvature flow . . . . .

26

Takashi Sakai
Antipodal sets of generalized s-manifolds . . . . . . . . . . . . . . . . . . . .

46

Kaname Hashimoto
Higher dimensional generalization of the Chiang Lagrangian and totally complex submanifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

62Masahiro Morimoto
Minimal PF submanifolds in Hilbert spaces with symmetries . . . . . . . . .

78

Makoto Kimura
Lagrangian submanifolds in complex projective space and quaternionic Kähler
geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
Hikaru Yamamoto
An introduction to the deformed Hermitian Yang-Mills (dHYM) connections

123

Hikaru YamamotoLine bundle mean curvature flows and the moduli space of dHYM connections 129
Akifumi Ochiai
Lagrangian mean curvature flows with generalized perpendicular symmetries . 135
Keita Kunikawa
Ricci flow, heat equation, Liouville type theorem . . . . . . . . . . . . . . . . 148
Makiko Sumi Tanaka
Antipodal sets of compact symmetric spaces and polars of compact Lie groups 157

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Ancient solutions for mean curvature flow of
isoparametric submanifolds
Xiaobo Liu
Ancient solutions are important in studying singularities of mean curvature flows(MCF). So far most rigidity results about ancient solutions are modeled on shrinking
spheres or spherical caps. In this talk, I will describe the behavior of MCF for aclass of submanifolds, called isoparametric submanifolds, which have more complicatedtopological type. We can show that all such solutions are in fact ancient solutions,i.e. they exist for all time which goes to negative infinity. I will also describe ourconjectures proposed together with Terng on rigidity of ancient solutions to MCF for
hypersurfaces in spheres. These conjectures are closely related to Chern’s conjecture
for minimal hypersurfaces in spheres. This talk is based on joint works with Chuu-Lian
Terng.
(X. Liu) Beijing International Center for Mathematical Research & School ofMathematical Sciences, Peking University, Beijing, China
E-mail address: xbliu@math.pku.edu.cn

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OCAMI Reports Vol. 2 (2021)

Ancient solutions for Mean
Curvature Flow of Isoparametric
Submanifolds
Xiaobo Liu
Peking University
Talk at Workshop on”Submanifolds of Symmetric Spaces and Their Time Evolution”
March 5-6, 2021

– p. 1/39

Mean curvature flow
Let M be a submanifold in a Riemannian
manifold X. The mean curvature flow
(abbreviated as MCF) of M is a map
f : I × M −→ X satisfying∂f= H(t, ·)∂twhere H(t, ·) is the mean curvature vector fieldof f (t, ·) and f (0, ·) is the immersion of M in X.

If a solution exists on I = (−∞, T ] for someT > 0, it is called an ancient solution. Suchsolutions are important in studying singularities
of general MCF.

– p. 2/39

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Examples:•If f0 : M −→ X is minimal, thenf (t, x) = f0 (x),x∈M

is an ancient solution to MCF. This is thestationary solution.
•

If M n ⊂ S n+1 is a subsphere, then the MCF of
M is an ancient solution, which shrinks to a point
in finite positive time and converges to theequator as t → −∞. This solution is called theshrinking spherical cap.

– p. 3/39•

If
f0 : M −→ S n+1 ⊂ Rn+2

is a minimal submanifold of the unit sphere S n+1
of any codimension (in particular M could be
S n+1 ), then√x∈Mf (t, x) = 1 − 2nt f0 (x),is an ancient solution to MCF for M as asubmanifold in Rn+2 .

– p. 4/39

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M ⊂ RN . νM : normal bundle of M .
Definition(Terng): M is isoparametric if(1) νM is globally flat, i.e. parallel translations ofnormal vectors along closed curves are identity.
(2) For any parallel normal vector field η,
principal curvatures along η are constant.
An isoparametric polynomial is a homogeneouspolynomial F on Rn+2 such that ∆F and ||∇F ||are constant along level sets of F .

– p. 5/39

Isoparametric hypersurfaces•

Let F be an isoparametric polynomial which isnormalized such that the range of F |S n+1 is[−1, 1]. Then for any t ∈ (−1, 1), F −1 (t) ∩ S n+1
is an isoparametric submanifold. These areisoparametric hypersurfaces in sphere.
M± := F −1 (±1) ∩ S n+1 are not isoparametric,
they are focal submanifolds of isoparametric
hypersurfaces.
Let g be the number of distinct principal
curvatures of an isoparametric hypersurface in
sphere. Then g = 1, 2, 3, 4, 6 (Münzner).

– p. 6/39

Submanifolds of Symmetric Spaces and Their Time Evolutions•

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Clifford examples(Ferus-Karcher-Münzner):
Assume that E1 , . . . , Em−1 are skew symmetricl × l matrices such thatEi Ej + Ej Ei = −2δij Id(i.e. these matrices give a representation of the
Clifford algebra.)
Then we can construct a homogeneouspolynomial of degree 4 on R2l which isisoparametric.
The corresponding isoparametric hypersurfaces
have 4 distinct principal curvatures. Most of them
are non-homogeneous.
– p. 7/39•

Homogeneous isoparametric submanifolds:
Let G/K be a symmetric space,g the Lie algebra of G,
k the Lie algebra of K,
p the orthogonal complement of k in g.
The isotropy representation of K acts on p.
Principal orbits of this representation areisoparametric submanifolds. All thesesubmanifolds are homogeneous.

•

Reducible cases: Products of isoparametricsubmanifolds are isoparametric.
– p. 8/39

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OCAMI Reports Vol. 2 (2021)

Properties•

If M ⊂ RN is complete isoparametric, thenM = M1 × Rk with M1 compact isoparametric.

If M ⊂ RN is compact isoparametric, then M is
contained in a round sphere. After translation anddilation, we may assume M is contained in theunit sphere centered at origin.
• For any parallel normal vector field η along
M ⊂ RN , define•Mη := {x + η(x) | x ∈ M }.
If M is isoparametric, then Mη is always asmooth submanifold.
– p. 9/39

Isoparametric foliation:
If dim Mη = dim M , then Mη is also
isoparametric. It is called a parallel
isoparametric submanifold of M .
• If dim Mη < dim M , then Mη is no longer
isoparametric. It is a focal submanifold of M .
SN•η Mη gives a singular foliation of R . If
M ⊂ S N −1 , then the set of Mη ⊂ S N −1 also
gives a singular foliation of S N −1 . These
foliations are called isoparametric foliations.
•– p. 10/39

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MCF of Isop. Submanifolds
Assume M ⊂ S N −1 ⊂ RN compact
isoparametric. MCF of M as a submanifold of
RN (resp. S N −1 ) is called the Euclidean (resp.
spherical) MCF of M .
Theorem(Liu-Terng, Duke 2009):• MCFs of M preserve the isoparametric condition
before collapsing.
• Euclidean MCF of M always converge to a focal
submanifold in a finite time T > 0. Same is true
for spherical MCF if M is not minimal in sphere.
• Every focal submanifold is a limit of the MCF ofsome isoparametric submanifold.
– p. 11/39

The above results were generalized:
for MCF of equifocal submanifolds of symmetricspaces by N. Koike (2011),
and for MCF of regular leaves of isoparametricfoliations on compact non-negativelt curved
manifolds by Alexandrino-Radeschi (2016).

– p. 12/39

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Ancient Solutions
Theorem(Liu-Terng, Math. Ann. 2020):
In both Euclidean and spherical cases,• MCF of M are ancient solutions, i.e. they exist
for all t ∈ (−∞, 0].
• There is a unique minimal isoparametricsubmanifold Mmin for each isoparametricfoliation in S N −1 .
• As t → −∞, MCF of M converges to MCF of
Mmin . (Note that the spherical MCF of Mmin isstationary).

– p. 13/39

More precisely, ∃ a unit parallel normal vector
field ζ on M in S N −1 such that the map
h : M → S N −1 defined byh(x) = (cos r)x + (sin r)ζ(x)is the embedding of Mmin in S N −1 , where r is thespherical distance between M and Mmin .
Let f (t, x), F (t, x) be the spherical andEuclidean MCF of M . For all x ∈ M ,√lim ||F (t, x) − 1 − 2nt h(x)|| = 0,t→−∞lim ||f (t, x) − h(x)|| = 0.

t→−∞

– p. 14/39

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Hypersurface cases
Let M n ⊂ S n+1 be an isoparametric hypersurface
with g distinct principal curvatures. For any
x0 ∈ M , let S 1 (x0 ) be the geodesic in S n+1
which passes x0 and is perpendicular to M .
Let M± be the focal submanifolds of M with
dim M+ ≤ dim M− . Let
m1 := dim M − dim M− ,m2 := dim M − dim M+ .
Then m1 and m2 are multiplicities of principal
curvatures of M .
Note that m1 ≤ m2 .
If g = 1, 3, 6, then m1 = m2 .
– p. 15/39

M+ ∪ M− intersects S 1 (x0 ) in exactly 2g points,
evenly distributed along the circle. Let x± be theintersection of M± with S 1 (x0 ) which are closest
to x0 . We may identify the normal space of M in
Rn+2 at x0 with C such that S 1 (x0 ) is the unit
circle in C and x+ = 1, x− = eiπ/g , x0 = eiθ0with 0 < θ0 < π/g.
Every parallel isoparametric hypersurface
intersect the arc {eiθ | 0 < θ < π/g} at exactly
one point. We can use this point to represent theparallel isoparametric hypersurface.

– p. 16/39

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OCAMI Reports Vol. 2 (2021)

Let Mmin be the unique minimal isoparametric
hypersurface which is parallel to M . Then Mminis represented by eiθmin wherecos gθmin = −δwithm2 − m1≥ 0.
m2 + m1
Theorem (Liu-Terng): Let Mt be the sphericalmean curvature flow of M . Assume Mt isrepresented by eiθ(t) . Then θ(t) is given byδ :=cos gθ(t) = egnt (cos gθ0 + δ) − δ.
– p. 17/39

Note that eiθmin divides the arc{eiθ | 0 < θ < π/g}
into two parts. θ(t) → 0 or π/g as t approaches
some positive number. As t → −∞, θ(t) → θmin .
Remark: For higher codimensional cases,
explicit solutions of MCF can also be constructedrecursively using Coxeter group structure of theisoparametric submanifolds. In general, it is moredifficult to locate the position of minimal leaf of
isoparametric foliation in higher codimension.
– p. 18/39

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Rigidity Results
Theorem (Huisken and Sinestrari, JDG 2015):
Let Mt be the spherical MCF of a hypersurface inS n+1 , A(t) and H(t) be the shape operator andthe mean curvature vector field of Mt as asubmanifold of S n+1 . Mt is either a shrinking
spherical cap or a stationary solution if one of thefollowing conditions is satisfied for all t < 0:
• (HS1) For n ≥ 3 and||A(t)||2 −•1||H(t)||2 ≤ 2.
n−1(HS2) For some constant B < 4n,||A(t)||2 < e−Bt ||H(t)||2 .

– p. 19/39

Huisken and Sinestrari claimed that condition(HS1) is sharp. They justify this claim byconsidering the MCF of a product of an(n − 1)-dimensional sphere and a circle in S n+1 .
This is precisely the MCF of an isoparametric
hypersurface with g = 2, m1 = 1, m2 = n − 1.
According to our calculations, for this example||A(t)||2 −n−2
1||H(t)||2 − 2 =tan2 θ(t)n−1n−1where eiθ(t) represents Mt . As t → −∞,θ(t) → θmin , RHS → n − 2 which is notarbitrarily small. So the justification for the
sharpness of (HS1) is not correct.

– p. 20/39

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Lemma(Liu-Terng): For every g = 1, 2, 3, 4, 6, ∃isoparametric hypersurface with g distinct
principal curvatures such that its MCF satisfies||A(t)||2 < e−2gnt ||H(t)||2
for all t < 0.
Corollary: Condition (HS2) is sharp in the sensethat B can not be ≥ 4n. (Otherwise g = 2 case of
the above lemma would give counter examples.)– p. 21/39

Rigidity conjectures
Conjecture A(Liu-Terng): If ∃ c1 , c2 , T > 0 suchthat||A(t)||2
−2gnt
−2gnt
≤
c
e
c1 e
≤
2
||H(t)||2

for all t < −T , then the ancient solution of MCF
Mt of compact hypersurfaces in sphere is the
MCF of an isoparametric hypersurface with g
distinct principal curvatures.
Remark: g = 1 case of this conjecture is true by
Huisken and Sinestrari’s result.
The above estimate holds for MCF of
isoparametric hypersurfaces.

– p. 22/39

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Conjecture B(Liu-Terng): If ∃ some constants
0 < ǫ < 1 and T > 0 such that either
1(g − 1)n ≤ ||A(t)||2 − ||H(t)||2 ≤ (g − 1 + ǫ)nnor1(g − 1 − ǫ)n ≤ ||A(t)||2 − ||H(t)||2 ≤ (g − 1)nnfor all t < −T , then the ancient solution of MCF
Mt of compact hypersurfaces in sphere is the
MCF of an isoparametric hypersurface with g
distinct principal curvatures.
– p. 23/39

Remark: g = 1 case of Conjecture B is true by aresult of Lei-Xu-Zhao (2019).
The above estimate holds for MCF of
isoparametric hypersurfaces.
The two inequalities in Conjecture B can not be
replaced by the following inequality:1(g−1−ǫ)n ≤ ||A(t)||2 − ||H(t)||2 ≤ (g−1+ǫ)n.
n
Otherwise Otsuki’s construction would give a
counter example (which are minimal of
topological type S n−1 × S 1 ).
– p. 24/39

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OCAMI Reports Vol. 2 (2021)

Chern’s Conjecture
Original Chern’s Conjecture: M n ⊂ S n+1
compact minimal with constant ||A||. Then theset of possible values of ||A||2 is discrete.
Remark: For minimal isoparametric
hypersurface with g distinct principal curvatures,||A||2 = n(g − 1).
Stronger version of Chern’s Conjecture:
M n ⊂ S n+1 compact minimal with constant||A||. Then M is isoparametric.

– p. 25/39

Stationary case of Conjecture B
Conjecture C: M n ⊂ S n+1 compact minimal. If∃ some constants 0 < ǫ < 1 such that either(g − 1)n ≤ ||A||2 ≤ (g − 1 + ǫ)nor(g − 1 − ǫ)n ≤ ||A||2 ≤ (g − 1)n,then M is an isoparametric hypersurface with g
distinct principal curvatures.
Remark: Conjecture C is stronger than Chern’s
conjecture.

– p. 26/39

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Theorem (Chern-do Carmo-Kobayashi, 1970):
M n ⊂ S n+1 compact minimal. If
0 ≤ ||A||2 ≤ n,then M is either an equator (isoparametric with
g = 1) or a Clifford torus (isoparametric with
g = 2).
This result implies that Conjecture C is true forg = 1 case and half of the g = 2 case.

– p. 27/39

Theorem: Assume M n ⊂ S n+1 compact
minimal withn ≤ ||A||2 ≤ (1 + ǫ)n.
M must be a Clifford torus if• ǫ = 1 and ||A|| is constant (Peng-Terng, 1983),12••ǫ = 13 and ||A|| is constant (Cheng-Yang, 1998),

ǫ = 37 and ||A|| is constant (Suh-Yang, 2007), or•1
ǫ = 23(Ding-Xin, 2011), or•1
ǫ = 22(Xu-Xu, 2017).
– p. 28/39

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These results give some partial answers forsecond half of the g = 2 case of Conjecture C.
Chern’s conjecture for isoparametric
hypersurface in S 4 was proved by S.P. Chang(1993).
The above results was proved using estimatesobtained from elliptic equations for ∆∐ and∆(∇∐). We hope the flow (parabolic) method
may provide new insights to Chern’s conjecture.

– p. 29/39

Idea for Proof: Euclidean Case
Assume M ⊂ S N −1 ⊂ RN compact
isoparametric. Fix p ∈ M . Let V = νp M be thenormal space of M as a submanifold in RN at p.
The intersection of V and the union of all focal
submanifolds of M is a union of finitely many
hyperplanes in V . Let W be the group generated
by reflections along these hyperplanes.
W is a finite group, called the Coxeter group of
M (Terng).

– p. 30/39

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Let C be the interior of the fundamental domainof W acting on V which contains p. C is called
the open Weyl chamber of W .
C is an open simplicial cone in V .
Every parallel isoparametric submanifold of M
intersect C at exactly one point.
Every focal submanifold of M intersect ∂C atexactly one point.

– p. 31/39

MCF of M is reduced to a flow equation forpoints in C ⊂ V .
W invariant polynomials on V give a newcoordinate system on C. In this coordinates, theEuclidean MCF becomes a flow equation along apolynomial vector field, and can be solved
recursively in a rather explicit way. The solution
exists as long as it does not hit ∂C.

– p. 32/39

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Since focal submanifolds have lower dimensions,the volume function approaches 0 if the flow
approaches ∂C.
As t → −∞, the volume function of MCF isincreasing, hence never hits ∂C. Consequently
MCF exists for all t ∈ (−∞, 0], i.e. the solution
is ancient.

– p. 33/39

By a result of Palais-Terng, each isoparametricfoliation in S N −1 contains at least one minimal
isoparametric leaf.
To study the limit of MCF as t → −∞, let x(t) be
the Euclidean MCF of M and x̃(t) the Euclidean
MCF of a minimal isoparametric submanifold in
S N −1 which is parallel to M . DefineD(t) := ||x(t) − x̃(t)||2 .

– p. 34/39

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Lemma: ∃ constant b > 0 such thatD′ (t) ≥b
D(t).
1 − 2nt

Corollary: For all t < 0,b

D(t) ≤ D(0)(1 − 2nt)− 2n .
In particular, limt→−∞ D(t) = 0, i.e. MCF of M
converges to the MCF of a minimal isoparametricsubmanifold.
Corollary: Each isoparametric foliation of S N −1
contains exactly one minimal isoparametric leaf.

– p. 35/39

Spherical Case
Let f (t, x) and F (t, x) denote the spherical andEuclidean MCF of M . Then√1
F (t, x) = 1 − 2nt f (− ln(1 − 2nt), x).
2nWe can use this formula to prove results about
spherical MCF of M .

– p. 36/39

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Geometric quantities
Let αi be unit normal vectors of walls of Coxeter
group (roots), mi multiplicities of curvature
distribution. For x ∈ C,
X mi αiE,
H (x) = −
hx,
α
i
i
i
X mi αinx
H S (x) = −
,
+
2
hx,
α
i
kxk
i
i
X mi
kAE (x)k2 =
2,
hx,
α
i
i
i
X mi
n
kAS (x)k2 =2 − kxk2 .
hx, αi ii– p. 37/39

If x(t) ∈ C represents a solution to MCF, thenlim (1 − 2nt) kH E (x(t))k2 = n2 ,
t→−∞
X
E
2
lim (1 − 2nt) kA (x(t))k =
t→−∞

i

mi
.
hxmin , αi i2andlim kH S (y(t))k2 = 0,
t→−∞
X
S
2
lim kA (y(t))k =
t→−∞

i

mi
− n.
hxmin , αi i2– p. 38/39

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Thanks!

– p. 39/39

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OCAMI Reports Vol. 2 (2021)

Polar foliations in symmetric spaces and their mean
curvature flow
Xiaobo Liu
Polar foliations are natural generalizations of orbit foliations of polar actions. In
this talk I will describe the relation between polar foliations and isoparametric submanifolds in simply connected symmetric spaces with non-negative curvature. It turns out
principal orbits of such foliation are isoparametric submanifolds. If leaves are compact,
such foliations must be products of isoparametric foliations in Euclidean spaces andpolar foliations in compact symmetric spaces. For polar foliation in compact symmetric spaces, there is a unique regular leaf which is minimal. The mean curvature flowof all regular leaves have ancient solutions and always converge to the minimal regularleaf as time goes to negative infinity. This talk is based on joint works with Marco
Radeschi.
(X. Liu) Beijing International Center for Mathematical Research & School ofMathematical Sciences, Peking University, Beijing, China
E-mail address: xbliu@math.pku.edu.cn

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Polar Foliations on SymmetricSpaces and Mean Curvature Flow
Xiaobo Liu
Peking University
Talk at Workshop on”Submanifolds of Symmetric Spaces and Their Time Evolution”
March 5-6, 2021

– p. 1/37

Isoparametric Submanifolds
Let N be a Riemannian manifold, M ⊂ N asubmanifold.
Given a parallel normal vector field ξ definedover a small open subset U ⊂ M , letUξ := {exp(ξ(p)) | p ∈ U }.
If kξk is small, Uξ is a smooth submanifold of N .
We call Uξ a locally parallel submanifold of M .

– p. 2/37

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Defination(Heintze-Liu-Olmos, 1997): A
submanifold M ⊂ N is isoparametric if(1) Normal bundle νM is flat.
(2) For every p ∈ M , exp νp M is totally geodesic
in a neighbourhood of p (called a local section).
(3) Locally parallel submanifolds of M haveparallel mean curvature vector fields.

– p. 3/37

Example: If N is a space form, this coincides
with Terng’s definition of isoparametricsubmanifolds.
Unlike the space form case, isoparametricsubmanifolds in general Riemannian manifold:(1) may not have constant principal curvatures.
(2) may not produce a global singular foliation of
N . If it does, such foliation is called an
isoparametric foliation.

– p. 4/37

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29

Example: If N is a compact symmetric space,M ⊂ N is an equifocal submanifold if(1) νM is flat and abelian (use Lie algebra
structure).
(2) Focal distance and multiplicities along locallyparallel normal vector fields are constant.
(This definition is due to Terng andThorbergsson)
Fact: M is equifocal if and only if it isisoparametric with flat sections.

– p. 5/37

Example: Let G be a Lie group acting
isometrically on a Riemannian manifold N . The
action is polar if there exists a totally geodesic
submanifold Σ ⊂ N which intersects all orbits of
G and intersects them orthogonally, and thedimension of Σ is complementary to thedimension of principal orbits. Σ is called asection of the polar foliation.
The action is hyperpolar if it is polar with flat
section.
Fact: Principal orbits of polar actions areisoparametric.
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Given an isoparametric foliation (N, F), i.e. a
singular foliation consists of an isoparametricsubmanifold L and all its parallel submanifolds.
It turns out that all regular leaves (i.e. leaves
with maximal dimension) of F are isoparametricsubmanifolds as well. Passing through every
point p, there is a totally geodesic submanifold
Σp (i.e. a section) which intersects all leaves
orthogonally.
A vector v ∈ T N is horizontal if it is tangent toa section. It is vertical if it is tangent to a leaf.

– p. 7/37

Mean curvature flow
Recall mean curvature flow (abbreviated as
MCF) of M over an interval I is a map
f : I × M −→ Nsatisfying∂f= H(t, ·)∂twhere H(t, ·) is the mean curvature vector fieldof f (t, ·) and f (0, ·) is the immersion of M in N .
If a solution exists on I = (−∞, T ] for someT > 0, it is called an ancient solution. Suchsolutions are important in studying singularities
of general MCF.
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Theorem(Liu-Terng, Duke 2009): If L is an
isoparametric submanifold in a Sphere or
Euclidean space. Then
(1) MCFs of L flows through leaves of theisoparametric foliation associated to L.
(2) Euclidean MCF of L always converge to a
singular leaf in a finite time T > 0. Same is true
for spherical MCF if L is not minimal in sphere.

– p. 9/37

The previous result was generalized:
for MCF of equifocal submanifolds in symmetricspaces by N. Koike (2011), andfor MCF of regular leaves of isoparametricfoliations on compact non-negatively curved
manifolds by Alexandrino-Radeschi (2016).

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Theorem(Liu-Terng, Math. Ann. 2020):
If L is an isoparametric submanifold in a Sphereor Euclidean space, then(1) MCF of L always has ancient solution, i.e. it
exists for all t ∈ (−∞, 0].
(2) There is a unique minimal isoparametricsubmanifold Lmin for each isoparametric foliationin S N −1 .
(3) As t → −∞, MCF of L converges to MCF of
Lmin . (Note that the spherical MCF of Mmin isstationary)– p. 11/37

Theorem A(Liu-Radeschi, 2020): Assume F is
an isoparametric foliation on a Riemannian
manifold M such that RicM (v) > RicΣ (v) for
any section Σ and v tangent to Σ, and the leaf
space M/F is compact. Then
(1) There is a unique minimal regular leaf Lmin in
F.
(2) For any regular leaf L in F, the MCF of L
always has ancient solution and it converges to
Lmin as t goes to −∞.
Remark:This result generalizes thecorresponding result of Liu-Terng in spheres.
– p. 12/37

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33

Polar Foliation
Let N be a complete Riemannian manifold. A
singular foliation F on N is Riemannian if every
geodesic perpendicular to a leaf at one point mustbe perpendicular to all leaves which it intersects.
Moreover, F is polar if for ∀p ∈ N , ∃ a totally
geodesic submanifold Σ ∋ p with dimension
complementary to the dimension of regularleaves such that Σ intersects all leaves of F
orthogonally. Σ is called a section of F.
Remark: No restriction for mean curvature ofleaves in F.

– p. 13/37

A hyperpolar foliation is a polar foliation withflat sections.
Remark: In general a polar foliation may not be
isoparametric.
Starting from an isoparametric submanifold, its
parallel submanifolds may not form a foliation.

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OCAMI Reports Vol. 2 (2021)

Example:Orbits of polar actions form a polar foliation.
Orbits of hyperpolar actions form a hyperpolar
foliation.
Isoparametric foliations are polar.
Parallel submanifolds of an equifocal submifold
in compact simply connected symmetric spaces
form a hyperpolar foliation.

– p. 15/37

Theorem B(Liu-Radeschi, 2020): Let F be apolar foliation on a simply connected symmetricspace N with non-negative curvature. Then
(1) F is always isoparametric.
(2) F is a product of a polar foliation withcompact leaf space N/F and an isoparametricfoliation in a Eucliean space.
(3) If F has compact leaves, then it is a productof a polar foliation in a compact symmetric space
and an isoparametric foliation in a Eucliean
space.
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Corollary (Liu-Radeschi, 2020):
Let N be a simply connected symmetric spacewith curvature ≥ 0 and F is a polar foliation on
N without trivial or Euclidean factors. Then
(1) There is a unique regular leaf Lmin in F
which is minimal in N .
(2) MCF of every regular leaf of F has ancientsolution and it converges to Lmin as t 7→ −∞.
Remark: Together with Liu-Terng’s result for
isoparametric submanifolds in Euclidean spaces,
this result completeluy describes behavior of
MCF for polar foliation in nen-negatively curved
symmetric spaces in negative time direction.
– p. 17/37

Idea for proof of Theorem A
Let L be a regular leaf in a polar foliation F on a
Riemannian manifold N . Let γ(t) be a geodesic
with γ(0) = p ∈ L and γ ′ (0) ⊥ Tp L. Let X be aparallel normal vector field along L with
X(p) = γ ′ (0).
Define end-point map φX : L −→ N by
φX (q) := expq X(q).
For every v ∈ Tp L, we call Jv (t) := dp φtX (v) aholonomy Jacobi vector field along γ. Let Lt be
the leaf of F passing γ(t). Then Jv (t) ∈ Tγ(t) Lt
for all t.
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Let Σ ∋ p be a section of F. Then γ(t) ∈ Σ forall t.
Let Vt := νγ(t) Σ. We have
Vt ⊇ Tγ(t) Lt
for all t with "=" iff Lt is a regular leaf.
The curvature operator R on N defines asymmetric operator Rt on Vt by
Rt (w) := R(w, γ ′ (t))γ ′ (t)for w ∈ Vt .
– p. 19/37

There is a Raccati operator St on Vt such that
St (J(t)) = J ′ (t) for holonomy Jacobi vector field
J at regular times. In fact −St is the shapeoperator of Lt along γ ′ (t) if Lt is a regular leaf.
St satisfies the Riccati equation:
St′ + St2 + Rt = 0.
Riccati comparison theorem: Let n = dim Vt .
If n1 tr(Rt ) > δ > 0, then n1 tr(St ) is bounded
above by the solution ofs′ (t) + s2 (t) + δ = 0
with s(0) = n1 tr(S0 ).

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Submanifolds of Symmetric Spaces and Their Time Evolutions

37

Given an orthonormal basis {ei | i = 1, · · · , n}of Tp L where p = γ(0). Let Ji be the holonomy
Jacobi field with Ji (0) = ei . Let ω and ωt be thevolume element of L and Lt respectively. Then atp,Φ∗tX ωt = fp (t) · ω
where
fp (t) := det(< Ji (t), Ej (t) >)1≤i,j≤nwith Ej (t) the parallel extension of ej along γ.

– p. 21/37

Fact:
dln fp (t) = tr(St ) = − < Ht , Xt > .
dt
So we can use Riccati comparison theorem toestimate volume of Lt . In particular, we have
Lemma: If for all regular leaves L and allv ∈ νp L, trTp L R(·, v)v > 0, then1
V (L) := vol(L) n is a strictly concave function on
the regular part of N/F.
Corollary: If in addition, N/F is compact, thenthere is a unique minimal regular leaf Lmin .
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OCAMI Reports Vol. 2 (2021)

If F is isoparametric, MCF of a regular leaf canbe reduced to a flow on regular part of N/F and
V is a Lyapunov function for the flow along −H
with unique global attractor Lmin . Hence MCF of
L converges to Lmin as t 7→ −∞. This finishes
the proof of Theorem A.

– p. 23/37

Idea for proof of Theorem B
To prove a polar foliation F on a simply
connected manifold N is isoparametric, we need:
Theorem(Alexandrino-Toeben): Let L be a
regular leaf of F. Then
(1) L has trivial normal holonomy.
(2) L has constant focal data, i.e., the end point
map of a parallel normal vector field has constantrank.
So we only need to show mean curvature vector
fields along regular leaves are parallel.

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39

Let N be a simply connected symmetric spacewith curvature ≥ 0.
Theorem(Lytchak): Every polar foliation in N isa product of a trivial factor, a hyperpolar foliation,and polar foliations with spherical sections (i.e.
sections with constant positive curvature).
Theorem(Heintze-Liu-Olmos): Hyperpolar
foliations on N are isoparametric.
So we only need to show polar foliations withspherical sections are isoparametric.
– p. 25/37

We may assume every section Σ is a sphere withcurvature = 1 after re-scaling metric on N . Fix aregular leaf L of F. Let X be a unit parallel
normal vector field along L. For any p ∈ L, letγ(t) be the geodesic with γ(0) = p andγ ′ (0) = X(p). Then γ is periodic with period 2π.
In fact expq tX(q) is periodic for all q ∈ L. Thisimplies that all holonomy Jacobi fields are
periodic with period 2π.

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We can prove:(1) Periodicity of holonomy Jacobi fields implies
that eigenvalues of Rt along γ(t) are squares ofintegers
0 ≤ λ21 ≤ · · · ≤ λ2n .
P
Let m = ni=1 λi . Then 2m is the index of spaceof holonomy Jacobi fields along γ|[0,2π) . By
continuity of index, we see m is independent ofp ∈ L.

– p. 27/37(2) Let {e1 , . . . , en } be an orthonormal basis of
V0 where ei is an eigenvector of R0 witheigenvalue λ2i . Let Ji (t) be the holonomy Jacobi
vector field with Ji (0) = ei . Let Ei (t) be theparallel translation of ei along γ(t). Let
fp (t) := det(< Ji (t), Ej (t) >).
Then
fp (t) =Xai sin(si t) + bi cos(si t)iwhere si = λ1 ± λ2 ± · · · ± λn .
– p. 28/37

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41(3) It follows that fp lies in a space T offunctions with dim T = m + 1.
(4) Assume γ intersects singular leaves Lj at time
tj ∈ (0, π) for j = 1, . . . , k. LetP
mj := dim(L) − dim(Lj ). Then kj=1 mj = m
andfp(d) (tj ) = 0
for all d = 0, . . . , mj − 1 and j = 1, . . . , k. This
gives a system of m linearly independentconditions on space T which, together withcondition fp(0) = 1, uniquely determines fp .
– p. 29/37(5) Note that tj and mj are determined by focal
data of exp tX : L −→ N . By constancy of focal
data, fp (t) does not depend on p ∈ L.
(6) Sincefp′ (0) = fp (0)tr(S0 ) = − < Hp , Xp >,it follows that < Hp , Xp > does not depend onp ∈ L for all parallel normal vector field X. So
H is parallel along all regular leaves and F isisoparametric.

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OCAMI Reports Vol. 2 (2021)

Second part of Theorem B is a splitting theorem
for polar foliation. We would like to show thatafter splits off Euclidean factors, either N/F iscompact, or N is compact.
It follows from a combination of Lytchak’s
results that if (N, F) is an indecomposable polar
foliation with spherical sections, than N has to becompact. So we only need to consider thehyperpolar case.

– p. 31/37

Assume F is a hyperpolar foliation on M . For p
in the regular part of the foliation, let Lp be theleaf through p. DefineD(p) := {v ∈ νp L | R(·, v)v ≡ 0}.
Since M is a symmetric space with non-netative
Ricci curvature, D(p) is always a vector spaceand defines an integrable totally geodesic
distribution when p varies in the regular part of
the foliation.

– p. 32/37

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43

Moreover, it follows that v ∈ D(p) is equvalent
to trR(·, v)v = 0. It is also equivalent to thegeodesic exp(tv) intersecting singular leaves
only finitely many times. The latter fact is provedby estimating index of the space of holonomy
Jacobi fields using Riccati comparison theorem.
A section Σ of F then splits as a product of two
submanifolds Σ1 × Σ2 , with Σ1 an integral
submanifold of D and Σ2 perpendicular to D.

– p. 33/37

Each hyperpolar foliation has a Weyl group W
which is generated by reflections along affine
hyperplanes of Σ called walls of W . One canshow that the above splitting of Σ induce a
splitting of W as well. To prove this, we need touse the fact that for any horizontal geodesic γwith γ(0) ∈ Σ1 ∩ Σ2 , it intersects walls of Weylgroup W either finitely times when γ ⊂ Σ1 or
infinitely many times when γ " Σ1 . So walls of
W is divided into disjoint union of two sets, one
has normal vectors tangent to Σ1 , another has
normal vectors tangent to Σ2 . The decompositionof walls gives decomposition of W .
– p. 34/37

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OCAMI Reports Vol. 2 (2021)

Decomposition of the Weyl group implies the
splitting of (N, F) into the product of two
foliations (M1 , F1 ) × (M2 , F2 ). (Terng,
Heintze-Liu, Ewert, Silva-Speranca)
The first factor (M1 , F1 ), which has Σ1 as asection, is an isoparametric foliation in Euclidean
space (possibly times a trivial factor). The second
factor (M2 , F2 ), which has Σ2 as a section, has
the property that its leaf space M2 /F2 is compact.
Every horizontal geodesic γ(t) in M2 meets
singular leaves infinitely many times in positive t
direction.
– p. 35/37

If leaves of (N, F) are compact, we can prove
M2 is compact by contradiction:
In fact, if M2 is not compact, we have a splitting
M2 = M ′ × Rk for some k > 0. We then show
the projection from a regular leaf L2 of F2 to Rk
is a submersion. This is not possible if L2 iscompact.
This completes the proof of Theorem B.

– p. 36/37

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Thanks!

– p. 37/37

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OCAMI Reports Vol. 2 (2021)

Antipodal sets of generalized s-manifolds
Takashi Sakai
Abstract. We introduce the notion of generalized s-manifold as a generalizationof symmetric spaces. Then we study maximal antipodal sets of generalized smanifolds. This is partly joint work with S. Ohno and Y. Terauchi.

1

Generalized s-manifolds

A Riemannian symmetric space M is a Riemannian manifold which has a geodesic
symmetry sx at each point x ∈ M . The family {sx }x∈M of geodesic symmetries satisfies
sx ◦ sy = ssx (y) ◦ sx

(1)

for all x, y ∈ M , that is, a Riemannian manifold M is a quandle. By generalizing (1)to (2), we define generalized s-manifolds as follows.
Definition 1. Let M be a smooth manifold and Γ a group. Let {ϕx }x∈M be a family ofgroup homomorphisms ϕx : Γ → Diff(M ) from Γ to the diffeomorphism group Diff(M )of M . Then (Γ, {ϕx }x∈M ) is called a generalized s-structure on M if it satisfies thefollowing conditions:
1. For each γ ∈ Γ, the map µγ : M × M → M ; (x, y) 7→ ϕx (γ)(y) is smooth.
(When Γ is a Lie group, µ : Γ × M × M → M ; (γ, x, y) 7→ ϕx (γ)(y) is smooth.)
2. For each x ∈ M , x is an isolated fixed point of the action of ϕx (Γ) on M , i.e.,
x is isolated in F (ϕx (Γ), M ) := {y ∈ M | ϕx (γ)(y) = y (∀γ ∈ Γ)}.
3. For any x, y ∈ M and γ, δ ∈ Γ,
ϕx (γ) ◦ ϕy (δ) ◦ ϕx (γ)−1 = ϕϕx (γ)(y) (γδγ −1 )(2)holds.
When Γ = Z2 , a generalized s-manifold is just a symmetric space in the sense of
Loos and Nagano. More generally, a regular s-manifold is a generalized s-manifoldwith Γ = Z, in particular a k-symmetric space is a generalized s-manifold with Γ = Zk
(cf. [2]). A Γ-symmetric space introduced by Lutz [3] is a generalized s-manifold witha finite abelian group Γ.
Let K be R, C or H. For n, n1 , . . . , nr ∈ N satisfying n1 + · · · + nr < n, we consider
a flag manifold
Fn1 ,...,nr (Kn ) := {x = (V1 , . . . , Vr ) | V1 ⊂ · · · ⊂ Vr ⊂ Kn , dim Vi = n1 + · · · + ni } .
This work was partly supported by the Grant-in-Aid for Science Research (C) No. 17K05223,
JSPS.

Submanifolds of Symmetric Spaces and Their Time Evolutions

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The flag manifold Fn1 ,...,nr (Kn ) has fibrations over Grassmannian manifolds. Symmetries of the Grassmannian manifolds induce a Γ-symmetric structure on Fn1 ,...,nr (Kn )with Γ = (Z2 )r . Furthermore, in [4], by using Γ-symmetric triples, we investigatedΓ-symmetric structures on R-spaces, which is a natural generalization of symmetric Rspaces. We gave a necessary and sufficient condition for an R-space to admit a natural
Γ-symmetric structure in terms of the root system. Then we classified R-spaces that
admit natural Γ-symmetric structures, when the root system is irreducible.

2

Antipodal sets of generalized s-manifolds

Chen and Nagano [1] studied antipodal sets of compact symmetric spaces. The definition of antipodal sets of compact symmetric spaces naturally extends to a generalizeds-manifold (M, Γ, {ϕx }x∈M ).
Definition 2. A subset A of M is called an antipodal set if ϕx (γ)(y) = y holds for all
x, y ∈ A and γ ∈ Γ, i.e., y ∈ F (ϕx (Γ), M ) := {y ∈ M | ϕx (γ)(y) = y (∀γ ∈ Γ)}. An
antipodal set A of M is said to be maximal if A = A0 holds for any antipodal set A0of M with A ⊂ A0 . The supremum of the cardinalities of antipodal sets of M , denotedby #Γ M , is called the antipodal number of M . An antipodal set A is said to be great
if its cardinality attains #Γ M .
Theorem 1 (Ohno-S.-Terauchi). Any maximal antipodal set of Fn1 ,...,nr (Kn ) with respect to the Γ-symmetric structure with Γ ∼= (Z2 )r is congruent to
A ={(hei1 , . . . , ein1 iK , hei1 , . . . , ein1 +n2 iK , . . . , hei1 , . . . , ein1 +···+nr iK )| 1 ≤ i1 < · · · < in1 ≤ n, 1 ≤ in1 +1 < · · · < in1 +n2 ≤ n, . . . ,1 ≤ in1 +···+nr−1 +1 < · · · < in1 +···+nr ≤ n,#{i1 , . . . , in1 +···+nr } = n1 + · · · + nr },where e1 , . . . , en is the standard basis of Kn .
Furthermore, in [4], we determined maximal antipodal sets of natural Γ-symmetricstructures on R-spaces. More precisely, an R-space M can be realized as an orbit of theisotropy representation (s-representation) of a compact symmetric space G/K. Then
any maximal antipodal set of the natural Γ-symmetric structure on M is given as anorbit of the Weyl group of G/K. Consequently we obtain that the antipodal number of
M is the cardinality of the orbit of the Weyl group, hence it is equal to dim H∗ (M ; Z2 ).
These are generalizations of results on maximal antipodal sets and the two-numbers of
symmetric R-spaces by Tanaka–Tasaki [6] and Takeuchi [5].
References[1] B.-Y. Chen, T. Nagano, A Riemannian geometric invariant and its applications to a problem of
Borel and Serre, Trans. Am. Math. Soc. 308 (1988), 273–297.
[2] O. Kowalski, Generalized symmetric spaces, Lecture Notes in Mathematics 805, Springer-Verlag,
Berlin-New York, 1980.

48

OCAMI Reports Vol. 2 (2021)[3] R. Lutz, Sur la géométrie des espaces Γ-symétriques, C. R. Acad. Sci., Paris, Sér. I 293 (1981),
55–58.
[4] P. Quast and T. Sakai, Natural Γ-symmetric structures on R-spaces, J. Math. Pures Appl. (9)141 (2020), 371–383.
[5] M. Takeuchi, Two-number of symmetric R-spaces, Nagoya Math. J. 115 (1989), 43–46.
[6] M. S. Tanaka, H. Tasaki, Antipodal sets of symmetric R-spaces, Osaka J. Math. 50 (2013),
161–169.
(T. Sakai) Department of Mathematical Sciences, Tokyo Metropolitan University,
1-1 Minami-Osawa, Hachioji-shi, Tokyo 192-0397, Japan
E-mail address: sakai-t@tmu.ac.jp

Submanifolds of Symmetric Spaces and Their Time Evolutions

Antipodal sets of generalized s-manifolds
Takashi Sakai(Tokyo Metropolitan University / OCAMI)

March 5, 2021
Submanifolds of symmetric spaces and their time evolutions

Antipodal sets of generalized s-manifolds

Introduction
Deﬁnition(M, g) : Riemannian symmetric spacedef⇐⇒ For each x ∈ M , there exists sx ∈ Isom(M, g) s.t.
1

s2x = idM ,2

x is isolated in F (sx , M ) := {y ∈ M | sx (y) = y}.

En , S n , RHn , Gr (Kn ), compact Lie groups, ...
E. Cartan classiﬁed Riemannian symmetric spaces.
Generalizations:e.g.

locally symmetric space ∇R = 0
pseudo-Riemannian symmetric space, aﬃne symmetric space
weakly symmetric space (Selberg)
k-symmetric space, s-manifold (Ledger, Obata, Kowalski, ...)
Γ-symmetric space (Lutz)
Antipodal sets of generalized s-manifolds

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Symmetric space
Deﬁnition (Loos, Nagano)
M : symmetric spacedef⇐⇒ For each x ∈ M , there exists sx ∈ Diﬀ(M ) s.t.
1

µ : M × M → M ; (x, y) 7→ sx (y) is smooth,2s2x = idM ,3x is isolated in F (sx , M ) := {y ∈ M | sx (y) = y},4sx ◦ sy = ssx (y) ◦ sx(∀x, y ∈ M ).

Euclidean space Rnsx (y) = −y + 2x

sphere S n ⊂ Rn+1

sx (y) = −y + 2hx, yixe.g.

Antipodal sets of generalized s-manifolds

Γ-symmetric spaces
Deﬁnition (Lutz)
M : C ∞ -manifold, Γ : ﬁnite abelian groupµ = {µγ : M × M → M smooth map | γ ∈ Γ}
Γ-symmetric strucure on M
def⇐⇒
1 For each x ∈ M
Γ → Diﬀ(M ); γ 7→ γx := µγ (x, ·)is an injective homomorphism, i.e. Γx := {γx | γ ∈ Γ} ∼
= Γ.
2

3

3

Every x is isolated in
F (Γx , M ) := {y ∈ M | γx (y) = y (∀γ ∈ Γ)}.
For all x ∈ M and γ ∈ Γ, γx is an automorphism of µ, i.e.
µδ γx (y), γx (z) = γx µδ (y, z) (∀y, z ∈ M, δ ∈ Γ).
⇐⇒

δγx (y) ◦ γx = γx ◦ δy(∀x, y ∈ M, γ, δ ∈ Γ)
Antipodal sets of generalized s-manifolds

Submanifolds of Symmetric Spaces and Their Time Evolutions

Generalized s-manifolds
Deﬁnition (Ohno-S.)
M : C ∞ -manifold, Γ : (Lie) group(Γ, {φx }x∈M ) : generalized s-structure on M
def⇐⇒
1 For each x ∈ M , φ : Γ → Diﬀ(M ) is a groupx
homomorphism.
2

3

4

For each γ ∈ Γ, µγ : M × M → M ; (x, y) 7→ φx (γ)(y) is asmooth mapping. (In the case where Γ is a Lie group,µ : Γ × M × M → M ; (γ, x, y) 7→ φx (γ)(y) is smooth.)x is isolated in
F (φx (Γ), M ) := {y ∈ M | φx (γ)(y) = y (∀γ ∈ Γ)}.
For ∀x, y ∈ M and ∀γ, δ ∈ Γ
φx (γ) ◦ φy (δ) ◦ φx (γ)−1 = φφx (γ)(y) (γδγ −1 ).
Antipodal sets of generalized s-manifolds

Generalized s-manifolds

For each x ∈ M , Γx := {φx (γ) | γ ∈ Γ} is a subgroup of Diﬀ(M ).
We call Γx the symmetric transformation group at x ∈ M .

RemarkΓ = Z2=⇒(M, Γ, {φx }x∈M ) is a symmetric spaceΓ = Zk=⇒(M, Γ, {φx }x∈M ) is a k-symmetric space

Γ=Z

=⇒

(M, Γ, {φx }x∈M ) is a regular s-manifold

Γ is a ﬁnite abelian group=⇒ (M, Γ, {φx }x∈M ) is a Γ-symmetric space

Antipodal sets of generalized s-manifolds

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Antipodal sets of compact symmetric spaces
M : compact Riemannian symmetric space
Deﬁnition (B.-Y. Chen-Nagano)For x ∈ M , each connected component of the ﬁxed point set
F (sx , M ) := {y ∈ M | sx (y) = y}of sx is called a polar. An isolated polar is called a pole.
Deﬁnition (B.-Y. Chen-Nagano)
1

2

A ⊂ M : antipodal set
def⇐⇒ sx (y) = y for all x, y ∈ A
A ⊂ M : maximal antipodal set
def⇐⇒ A′ ⊂ M : antipodal set, A ⊂ A′ =⇒ A = A′

3

#2 M := sup{#A | A ⊂ M : antipodal} 2-number

4

A ⊂ M : great antipodal set

def

⇐⇒

#A = #2 M

Antipodal sets of generalized s-manifolds

Antipodal sets of compact symmetric spaces
Example:Rn+1

o = he1 i ∈ RP n
F (so , RP n ) = {o} ∪ {x ∈ RP n | x ⊥ o}∼
= {o} ∪ RP n−1
A = {hei i | i = 1, . . . , n + 1}

6
en+1







e1

#2 RP n = n + 1 = dim H∗ (RP n , Z2 )Theorem (Takeuchi)
M : symmetric R-space=⇒#2 M = dim H∗ (M, Z2 )Antipodal sets of generalized s-manifolds-

e2Submanifolds of Symmetric Spaces and Their Time Evolutions

Antipodal sets of generalized s-manifolds(Γ, {φx }x∈M ) : generalized s-structure on M .
Deﬁnition
For x ∈ M , a connected component of the ﬁxed point set
F (φx (Γ), M ) := {y ∈ M | φx (γ)(y) = y (∀γ ∈ Γ)}
is called a polar. An isolated polar is called a pole.
Deﬁnition
1 A ⊂ M : antipodal set
def⇐⇒ φx (γ)(y) = y for all x, y ∈ A and γ ∈ Γ
2A ⊂ M : maximal antipodal set
def⇐⇒ A′ ⊂ M : antipodal set, A ⊂ A′ =⇒ A = A′3

#Γ M := sup{#A | A ⊂ M : antipodal} antipodal number

4

A ⊂ M : great antipodal set

def

⇐⇒

#A = #Γ M

Antipodal sets of generalized s-manifolds

Flag manifolds
K = R, C or Hn, n1 , . . . , nr ∈ N satisfying n1 + · · · + nr < n
(
)
n
V
⊂
V
⊂
·
·
·
⊂
V
⊂
K
1
2
r
Fn1 ,...,nr (Kn ) := x = (V1 , . . . , Vr )dim Vi = n1 + · · · + ni (∀i)
For x = (V1 , . . . , Vr ) ∈ Fn1 ,...,nr (Kn ), deﬁnesVi = 2PVi − idKn : Kn → Kn(i = 1, . . . , r),where PVi : Kn → Vi is the orthogonal projection. ThensVi ∈ Diﬀ(M ),Hences2Vi = idM ,sVi ◦ sVj = sVj ◦ sVi .

(Z2 )r ∼= hsV1 , . . . sVr i ⊂ Diﬀ(M ).

Therefore Fn1 ,...,nr (Kn ) is a generalized s-manifold with Γ = (Z2 )r .
Antipodal sets of generalized s-manifolds

53

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OCAMI Reports Vol. 2 (2021)

Flag manifolds
Z2 × Z2 × Z2

Fn1 ,n2 ,n3 (Kn )(V1 , V2 , V3 )@?

@Fn1 ,n2 (Kn )Fn1 ,n2 +n3 (Kn )(V1 , V2 )(V1 , V3 )

?

Fn1

Q

Q 
Q

 Q
+

s
Q

(Kn )

V1

R
@

Fn1 +n2 ,n3 (Kn )(V2 , V3 )

Q



Q 
Q

 Q

+
Q
s

Fn1 +n2Z2 × Z2(Kn )?

Fn1 +n2 +n3 (Kn )

V2

Z2

V3

Antipodal sets of generalized s-manifolds

Maximal antipodal sets of ﬂag manifolds
Theorem (Ohno-S.-Terauchi)
1

Any maximal antipodal set of Fn1 ,...,nr (Kn ) with respect tothe Γ ∼= (Z2 )r -symmetric structure is congruent to

A ={(hei1 , . . . , ein1 iK , hei1 , . . . , ein1 +n2 iK , . . . , hei1 , . . . , ein1 +···+nr iK )| 1 ≤ i1 < · · · < in1 ≤ n, 1 ≤ in1 +1 < · · · < in1 +n2 ≤ n, . . . ,1 ≤ in1 +···+nr−1 +1 < · · · < in1 +···+nr ≤ n,#{i1 , . . . , in1 +···+nr } = n1 + · · · + nr },where e1 , . . . , en is the standard basis of Kn .
2

#Γ (Fn1 ,...,nr (Kn )) =n!
n1 !n2 ! · · · nr !nr+1 !

= dim H∗ (Fn1 ,...,nr (Kn ); Z2 ),where nr+1 := n − (n1 + · · · nr ).
Antipodal sets of generalized s-manifolds

Submanifolds of Symmetric Spaces and Their Time Evolutions

Kähler C-spaces
G : compact connected semisimple Lie groupx0 (6= 0) ∈ g
M:= Ad(G)x0 ⊂ g∼= G/Gx0
G x0: Kähler C-space:= {g ∈ G | Ad(g)x0 = x0 }Z(Gx0 ) := {g ∈ Gx0 | gh = hg (∀h ∈ Gx0 )}For each x = Ad(gx )x0 ∈ M and γ ∈ Γ, deﬁne
φx (γ) : M → M ; y 7→ Ad(gx γgx−1 )y,φx : Γ → Diﬀ(G/K); γ → φx (γ).
Then (Γ, {φx }x∈M ) is a G-equivariant generalized s-structure.
Moreover M has k-symmetric structures for ∀k ≥ ∃k0 ,i.e. generalized s-structures with Γ ∼= Zk .
Antipodal sets of generalized s-manifolds

Maximal antipodal sets of Kähler C-spaces
M = Ad(G)x0 : a Kähler C-space with Γ = Z(Gx0 )Proposition (Ikawa-Iriyeh-Okuda-S.-Tasaki)
12

For x, y ∈ M
y is antipodal to x⇐⇒

[x, y] = 0

A ⊂ M : maximal antipodal set=⇒ ∃t ⊂ g : maximal abelian subalgebras.t. A = M ∩ t.
Hence A is an orbit of the Weyl group of g with respect to t.
In particular, any maximal antipodal sets of M are congruent
with each other by G.

Antipodal sets of generalized s-manifolds

55

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OCAMI Reports Vol. 2 (2021)

Generalized s-structures on complex ﬂag manifolds
Fn1 ,...,nk (Cn ) is realized as an adjoint orbit of G = SU(n):Fn1 ,...,nr (Cn ) ∼= SU(n)/S(U(n1 ) × · · · × U(nr+1 )) = G/Gx0
Observationn1 F
n1 ,...,nr (C ) admits Γ-symmetric structures with
Γ = Zk (k ≥ k0 ),Γ = (Z2 )r ,2

Γ = Z(Gx0 ).

Maximal antipodal sets of Fn1 ,...,nr (Cn ) for these three
generalized s-structures coincide, that is an orbit of the Weylgroup of SU(n).

Antipodal sets of generalized s-manifolds

Compact symmetric triads(G, K1 , K2 ) : compact symmetric triad
i.e. (G, K1 , σ1 ), (G, K2 , σ2 ) : compact symmetric pairsΓ := hσ1 , σ2 i ⊂ Aut(G)Gπ?

G/(K1 ∩ K2 )π1
@ π2

Γ-symmetric space

R
@

G/K1
σ1 = σ2

=⇒

Γ∼
= Z2

σ1 6= σ2 , σ1 σ2 = σ2 σ1
σ1 σ2 6= σ2 σ1

G/K2

=⇒

=⇒

Γ∼
= Z2 × Z2

Γ = hσ1 , σ2 i is a non-abelian group
Antipodal sets of generalized s-manifolds

Submanifolds of Symmetric Spaces and Their Time Evolutions

Γ-symmetric triples
Deﬁnition (Lutz, Goze-Remm, Ohno-S.-Terauchi)(G, K, Γ) : Γ-symmetric triple
def⇐⇒ G : connected Lie groupΓ ⊂ Aut(G) : ﬁnite subgroup
K ⊂ G : closed subgroup s.t
F (Γ, G)0 ⊂ K ⊂ F (Γ, G),
where F (Γ, G) := {g ∈ G | γ(g) = g (∀γ ∈ Γ)}.
(G, K, Γ) : Γ-symmetric triple
For each x = gx K ∈ G/K and γ ∈ Γ, deﬁne
φx (γ) : G/K → G/K; gK 7→ gx γ(gx−1 g)K,
φx : Γ → Diﬀ(G/K); γ → φx (γ).
Then (Γ, {φx }x∈G/K ) is a G-equivariant generalized s-structure
on G/K.
Antipodal sets of generalized s-manifolds

Natural Γ-symmetric structures on R-spaces
P = G/K : simply-connected compact symmetric space
where G := Isom(P )0g=k⊕p
a ⊂ p : maximal abelian subalgebraR ⊂ a∗ : root system of (G, K) with respect to a
Σ := {α1 , . . . , αr } : simple roots of R
{ξ1 , . . . , ξr } : basis of a dual to Σ
For j ∈ {1, . . . , r}, deﬁne
gi := exp(πξi ) ∈ G,

and γ i := IntG (gi ) K ∈ Aut(K).

Note γ i γ i = idK , γ i 6= idK , γ i γ j = γ j γ i for all i, j ∈ {1, . . . , r},hence (Z2 )r ∼= hγ i : i = 1, . . . , ri ⊂ Aut(K).
Antipodal sets of generalized s-manifolds

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OCAMI Reports Vol. 2 (2021)

Natural Γ-symmetric structures on R-spaces
For I ⊂ Ireg := {1, . . . , r}, deﬁne ξI :=P

ξi andi∈I

XI := AdG (K)ξI ⊂ p R-space
XI ∼= K/HI where HI = {k ∈ K | AdG (k)ξI = ξI }.
Every R-space in p is equivariantly isomorphic to XI for someI ⊂ Ireg .
LemmaLet ∅ 6= I˜ ⊂ I ⊂ Ireg = {1, . . . , r}, then HI ⊂ HI˜.
Therefore the choice of I ⊂ Ireg gives a stratiﬁcation of orbit types.

Antipodal sets of generalized s-manifolds

Natural Γ-symmetric structures on R-spaces
ΓI = hγ i : i ∈ Ii ⊂ Aut(K)∼= (Z2 )|I|
A non-empty subset I of Ireg = {1, . . . , r} is called admissible if(K, HI , ΓI ) is a ΓI -symmetric triple, that is, if
F (ΓI , K)0 ⊂ HI ⊂ F (ΓI , K).
Then ΓI induces a ΓI -symmetric structure on XI ∼= K/HI .
Theorem (Quast-S.)
I ⊂ Ireg = {1, . . . , r} is admissiblerP⇐⇒ for all α =cαi αi ∈ R+i=1
∀i ∈ I : cαi even =⇒
∀i ∈ I : cαi = 0 .

Antipodal sets of generalized s-manifolds

Submanifolds of Symmetric Spaces and Their Time Evolutions

59

Extrinsically Γ-symmetric spaces
Assume that I ⊂ Ireg is admissible.
Then K/HI ∼= XI = Ad(K)ξI has a ΓI -symmetric structure.
K/HI → XI := AdG (K)ξI ; kHI 7→ AdG (k)ξI
γξiI (AdG (k)ξI ) = AdG (gi )AdG (k)ξI .
γξiI extends to the linear endomorphism AdG (gi )|p of p,
ΓIξI = hγξiI : i ∈ Ii ∼= hAdG (gi )|p : i ∈ Ii ⊂ End(p)F (ΓIξI , p) = {Y ∈ p | [Y, ξI ] = 0} = Tξ⊥i XI
Hnece XI can be considered as an extrinsically Γ-symmetricspace.
Antipodal sets of generalized s-manifolds

Maximal antipodal sets of natural Γ-structures on R-spaces
Let I be an admissible subset of Ireg = {1, . . . , r}.
K/HI → XI := AdG (K)ξI ;kHI 7→ AdG (k)ξIΓI -symmetric structure on K/HI can be transfered to XI as
γξiI (AdG (k)ξI ) = AdG (gi )AdG (k)ξI .
Hence γξiI extends to the linear endomorphism AdG (gi )|p of p.
F (ΓIξI , p) = {Y ∈ p | [Y, ξI ] = 0}
Theorem (Quast-S.)
For each antipodal subset A of XI , there exists some maximalabelian subspace a′ in p such thatA = XI ∩ a′ = W (P, a′ )ξI ,where W (P, a′ ) is the Weyl group of (G, K) with respect to a′ .
Any two maximal antipodal subsets of XI are conjugate by K.
Antipodal sets of generalized s-manifolds

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OCAMI Reports Vol. 2 (2021)

Maximal antipodal sets of natural Γ-structures on R-spaces

Corollary
Let A be a maximal antipodal subset of XI . Then#Γ (XI ) = |A| = W (P, a′ )ξI = dim H∗ (XI , Z2 ).

Antipodal sets of generalized s-manifolds

Further problems

1

Classify Γ-symmetric spaces and generalized s-manifolds.

2

Determine maximal antipodal sets and antipodal numbers ofgeneralized s-manifolds.

3

Study geometric meaning of the antipodal numbers ofgeneralized s-manifolds, in particular, a relationship with its
topology.

Thank you very much for your attention!

Antipodal sets of generalized s-manifolds

Submanifolds of Symmetric Spaces and Their Time Evolutions

References

B.-Y. Chen, T. Nagano, A Riemannian geometric invariant
and its applications to a problem of Borel and Serre, Trans.
Am. Math. Soc. 308 (1988), 273–297.
D. Ferus, Symmetric submanifolds of Euclidean space, Math.
Ann. 247 (1980), 81–93.
M. Goze, E. Remm, Riemannian Γ-symmetric spaces,
Diﬀerential geometry, 195–206, World Sci. Publ., Hackensack,NJ, 2009.
O. Loos, Symmetric spaces I, II, W. A. Benjamin, New York1969.

Antipodal sets of generalized s-manifolds

References

R. Lutz, Sur la géométrie des espaces Γ-symétriques, C. R.
Acad. Sci., Paris, Sér. I 293 (1981), 55–58.
P. Quast, T. Sakai, Natural Γ-symmetric structures on
R-spaces, J. Math. Pures Appl. 141 (2020), 371–383.
M. Takeuchi, Two-number of symmetric R-spaces, Nagoya
Math. J. 115 (1989), 43–46.
M. S. Tanaka, H. Tasaki, Antipodal sets of symmetricR-spaces, Osaka J. Math. 50 (2013), 161–169.

Antipodal sets of generalized s-manifolds

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OCAMI Reports Vol. 2 (2021)

Higher dimensional generalization of
the Chiang Lagrangian
and totally complex submanifolds
Jong Taek Cho, Kaname Hashimoto∗ , Yoshihiro Ohnita
Abstract. We will discuss a higher dimensional generalization of Chiang’s Lagrangian submanifold in CP 3 from the viewpoint of totally complex submanifoldsof HP n .

1

The Chiang Lagrangian

H2

S

The Chiang Lagrangian is a compact embedded minimal Lagrangian submanifold L3 of
CP 3 given as an SU (2)-orbit by the moment map method (R. Chiang [1]). It is known
to be a non-symmetric homogeneous space and a strictly Hamiltonian stable with nonparallel second fundamental form ([3], [5]), and so on ([6], [4]). The Chiang Lagrangian
L3 is envolved with several nice structures illustrated in the following diagram:πi

S 7 (1)
SO(4)/O(2)

πj

pi

π
Cj P

3

pj

S 4 (1)= HP 1S

π −1 (RP 2 )Ci P 3S3L̃3 = πj−1 (CP 1 )
SU (2)/Z3S
S 3 /Z3π

Z4L3 = πi (L̃)
hor. holom.

3

Ñ 2 = CP 1

Z2

Veronesemin. surf.
RP 2 = N 2

Here H = R1+Ri+Rj +Rk is the quaternion number field with quaternionic imaginary
units {i, j, k} and Hn+1 is considered as an (n+1)-dimensional quaternionic vecter spacewith the rightmultiplication of quaternionic numbers.

2

Totally complex submanifolds and R-spaces

We use the concept of “totally complex submanifolds” of a quaternionic projective space
HP n (e.g. [7]) in order to generalize the above diagram into the higher dimensionalsetting. A submanifold of HP n is called a totally complex submanifold if N can be∗

the presenterSubmanifolds of Symmetric Spaces and Their Time Evolutions

63

locally lifted to a twistor space CP 2n+1 over HP n as a complex Legendrian submanifold.
We have dimR N ≤ 2n. If dimR = 2n, then N is said to be of maximal dimension.
Main Theorem 1. The above diagram can be generalized into higher dimensions asfollows:S

Hn+1

S 4n+3 (1)πmin. Leg. j

Ci P 2n+1
S3

L̃2n+1 = πj−1 (Ñ )

pi

π
Cj P

2n+1

S

b = π −1 (N )
N

πi

pj
min. Lag.

HP n

L2n+1 = πi (L̃)
tot. cplx.
cplx. Leg.

S3
N 2n

Ñ n

π

Totally complex submanifolds of HP n with parallel second fundamental form havebeen classified by Kazumi Tsukada [7].
Theorem 1 (Tsukada [7]). Any n-dimensional totally complex submanifold Ñ in HP nwith parallel second fundamental form is locally congruent to one of canonically immersed totally complex submanifolds:(0) CP 1 −→ RP 2 ⊂ S 4 = HP 1
(1) CP n ⊂ HP n

(Veronese min surf.),(totally geodesic),(2) Sp(3)/U (3) −→ HP 6 ,(3) SU (6)/S(U (3) × U (3)) −→ HP 9 ,(4) SO(12)/U (6) −→ HP 15 ,(5) E7 /E6 · T 1 −→ HP 27 ,(6) CP 1 (c̃) × CP 1 (c̃/2) −→ HP 2 ,(7) CP 1 (c̃) × CP 1 (c̃) × CP 1 (c̃) −→ HP 3 ,(8) CP 1 (c̃) ×SO(n + 1)−→ HP nSO(2) × SO(n − 1)(n ≥ 4).

An R-space is a compact homogeneous space obtained as an orbit of the isotropy
representation of a Riemannian symmetric space, i.e. a so-called s-representaion. OlmosSánchez [8] showed the differential geometric characterization of R-spaces by means of

64

OCAMI Reports Vol. 2 (2021)

the parallelism of the second fundamental form with respect to a canonical connection. By constructing explicitly such a canonical connection on the inverse image
N̂ = π −1 (N ), we can obtain the following theorem.
Main Theorem 2. Assume that N is a 2n-dimensional totally complex submanifoldof HP n with parallel second fundamental form. Then its inverse image N̂ = π −1 (N ) ⊂S 4n+3 (1) ⊂ Hn+1 is a standardly embedded R-space associated to a quaternion-Kähler
symmetric pair (G, K).

totally cplx. imm. Ñ

q. K. symm. sp. G/KΠ(G, K)
G
2
HP 1
CP 1
G2
(Sp(1) × Sp(1))/Z2
SU (n + 3)
HP n
CP n (totally geodesic)
B2
S(U (2) × U (n + 1))
Sp(3)
F4
HP 6
F4
U (3)
(Sp(3) × Sp(1))/Z2
SU (6)E6
HP 9
F4
S(U (3) × U (3))(SU (6) × Sp(1))/Z2
SO(12)
E7
HP 15
F4
U (6)
(Spin(12) × SU (2))/Z2
E7
E8
HP 27
F4
(U (1) × E6 )/Z3
(E7 × SU (2))/Z2
SO(7)
HP 2
CP 1 (c̃) × CP 1 (c̃/2)(n = 2)
B3
SO(4) × SO(3)SO(8)(n = 3)HP 3
CP 1 (c̃) × CP 1 (c̃) × CP 1 (c̃)D4
SO(4) × SO(4)SO(n + 1)
SO(n + 5)
HP n CP 1 (c̃) ×(n ≥ 4)B4SO(2) × SO(n − 1)
SO(4) × SO(n + 1)
Here Π(G, K) denotes the Dynkin diagram of the restricted root systems of symmetricpair (G, K).
It gives a new proof of Theorem 1 different from [7].
G2In the case when G/K =, we have n = 1, N 2 is a Veronese(Sp(1) × Sp(1))/Z2
minimal surface of HP 1 = S 4 and L3 = πi (L̃) is nothing but the Chiang Lagrangian in
CP 3 .
Acknowledgments. The authors would like to thank Professor Naoyuki Koike for excellent organaization of the inernational workshop “Submanifolds of symmetric spaces
and thier time evolutions”. This work was partly supported by Osaka City University
Advanced Mathematical Institute: MEXT Joint Usage/Research Center on Mathematics and Theoretical Physics JPMXP0619217849.

Submanifolds of Symmetric Spaces and Their Time Evolutions

65

References[1] R. Chiang, New Lagrangian submanifolds of CP n . Int. Math. Res. Not. 45 (2004), 2437–2441.
[2] B. Y. Chen, F. Dillen, L. Verstraelen, and L. Vrancken, An exotic totally real minimal immersionsof S 3 in CP 3 and its characterization, Proc. Royal Soc. Edinburgh Sect. A, Math. 126 (1996),
153–165.
[3] L. Bedulli and A. Gori, A Hamiltonian stable minimal Lagrangian submanifold of projective spacewith nonparallel second fundamental form. Transform. Groups 12 (2007), no. 4, 611–617.
[4] J. D. Evans and Y. Lekili, Floer cohomology of the Chiang Lagrangian. Selecta Math. (N.S.) 21(2015), no. 4, 1361–1404.
[5] Y. Ohnita, Stability and rigidity of special Lagrangian cones over certain minimal Legendrian
orbits, Osaka J. Math. 44 no. 2 (2007), 305-334.
[6] Y. Ohnita, On deformation of 3-dimensional certain minimal Legendrian submanifolds, Proc.
The 13-th International Workshop on Differential Geometry and Related Fields, 13 (2009),
pp.71–87, NIMS, KMS and Grassmann Research Group.
[7] K. Tsukada, Parallel submanifolds in a quaternion projective space. Osaka J. Math. 22 (1985),
187–241.
[8] C. Olmos and C. Sánchez, A geometric characterization of the orbits of s-representations. J. reine
angew. Math. 420 (1991), 195–202.
[9] Y. Ohnita, Parallel Kähler submanifolds and R-spaces. (submitted), a preprint, OCAMI Preprint
Ser. 20-21.
[10] J.-T. Cho, K. Hashimoto and Y. Ohnita, Totally complex submanifolds and R-spaces. in preparation.
(Jong Taek Cho) Department of Mathematics, Chonnam National University, Gwangju
61186, Korea
E-mail address: jtcho@chonnam.ac.kr(K. Hashimoto) Osaka City University Advanced Mathematical Institute, 3-3-138
Sugimoto, Sumiyoshi-ku, Osaka, 558-8585, Japan & Yamato Univesity
E-mail address: h-kaname@sci.osaka-cu.ac.jp(Y. Ohnita) Osaka City University Advanced Mathematical Institute, & Department of Mathematics, Osaka City University, 3-3-138 Sugimoto, Sumiyoshi-ku, Osaka,558-8585, Japan
E-mail address: ohnita@sci.osaka-cu.ac.jp

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OCAMI Reports Vol. 2 (2021)

Higher dimensional generalization of
the Chiang Lagrangian
andtotally complex submanifolds
Kaname Hashimoto （OCAMI/Yamato Univ.)
Joint work with J. T. Cho (Chonnam Natl. U.), Y. Ohnita (OCAMI)
March 5 (Fri.), 2021
OCAMI workshop on“Submanifolds of Symmetric Spaces and their Time Evolutions”

R. Chiang Lagrangian submanifolds
Deﬁnition (IMRN, 2004)
The Chiang Lagrangian submanifold L3 ⊂ CP 3 is deﬁned by
(
)
2
2
2
2
3|z
|
+
|z
|
−
|z
|
−
3|z
|
=
0
0
1
2
3
L3 := [z0 : z1 : z2 : z3 ] ∈ CP 3z0 z¯1 + z1 z¯2 + z2 z¯3 = 0
minimal Lagrangian SU (2) orbit L3 = ρ(SU (2))[1 : 0 : 0 : 1].
conn. cpt. embedded minimal Lagranagian submanifold in CP 3 .
L3 does not possess parallel second fundamental forms. ∇∗ αN ̸= 0
Homogeneous space but not symmetric space.
Curvature characterization(B. Y. Chen, Dillen, Verstraelen, Vrancken, Bolton, 1996)
Strictly Hamiltonian stable (Ohnita, Bedulli-Gori, 2007)
Floer homology (Evans-Lekili, 2015), min.Maslov number of L3 = 2

Submanifolds of Symmetric Spaces and Their Time Evolutions

67

H = R + Ri + Rj + Rk
Ci := R + iR ⊂ H,

Cj := R + jR ⊂ H

We consider the following diagram:
H2S
S 7 (1)πj
Cj P 3πi
Ci P 3π S3pipj
S 4 (1)SH2πi

S 7 (1)πj

πCj P 3

2

(RP )

pj

S 4 (1) = HP 1

S

π

pi

S3

SO(4)/O(2)−1

Ci P 3L̃3 = πj−1 (CP 1 )

Z4

L3 = πi (L̃)
Veronesemin. surf.

hor. holom.

SU (2)/Z3
S3
S 3 /Z3

πÑ 2 = CP 1

Z2

RP 2 = N 2• CP 1 → Cj P 3 : Veronese embedding of degree 3.
• L̃3 ⊂ S 7 (1) : minimal Legendrian submanifold embedded in S 7 (1)68OCAMI Reports Vol. 2 (2021)

Higher dimensional generalization
Let Hn+1 ∼= R4n+4 be an (n + 1)-dimensional quaternionic vector spacewith right multiplications by i, j, k. We consider the following standard
ﬁbrations:
Hn+1
S
S 4n+1 (1)
πj

πi
S

1

S

Cj P 2n+1

1

Ci P 2n+1

π S3
S2

S2

pi

pj
HP n

Then Cj P 2n+1 has the standard complex contact structure and theholomorphic contact 1-form on Cj P 2n+1 .
Suppose that Ñ → Cj P 2n+1 is a horizontal holomorphic immersion of ann-dimensional complex manifolds Ñ , that is, a complex Legendrian
submanifold of Cj P 2n+1 .
Hn+1S
S 4n+1 (1)
πj

πi
S1

Cj P 2n+1Ñ
cplx. Leg.

S1
π S3

Ci P 2n+1

S2

S2

pi

pj
HP n

Submanifolds of Symmetric Spaces and Their Time Evolutions

69

Totally complex submanifold
Let (M 4n , g, Q) be a quaternionic Kähler manifold. g is the Riemannian
metric on M and Q is a rank 3 subbundle of End T M which satisﬁes thefollowing conditions:
For each p ∈ M , there is a neighborhood U of p over which there exists a
local frame ﬁeld {I, J, K} of Q satisfying
I 2 = J 2 = K 2 = −id,
IJ = −JI = K, JK = −KJ = I, KI = −IK = J.

Deﬁnition
A submanifold N 2k ⊂ (M 4n , g, Q) (k ≤ n) is said to be totally complex
if, for every p ∈ N , there exists an open neighborhood U of q in N andsections J of Q|U such that the following properties hold:(1) J(T U ) = T U ,(2) J 2 = −id,(3) I(Tq N ) ⊆ Tq⊥ N for every q ∈ U and I ∈ Qq such that Jq I = −IJq ,(4) ∇J = 0.
⇐⇒ N has cplx. Leg. submanifolds to twister spase Z.

A horizontal holomorphic map Ñ → Cj P 2n+1 of n-dimensional complex
manifold Ñ corresponds to a maximal dimensional totally complex
immersion Ñ → HP n .

S

Hn+1

πi

S 4n+3 (1)Ci P 2n+1

πj

πCj P 2n+1

min. Leg
L̃ = πj−1 (Ñ )HP nmin. Lag.

covering

L = πi (L̃)

cplx. Leg.

tot. cplx.

Ñ n

N 2n

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Theorem

S

Hn+1

πi

S 4n+3 (1)Ci P 2n+1
S3

πj

pi

π
Cj P 2n+1S

b = π −1 (N ) min. Leg.
N
L̃2n+1 = πj−1 (Ñ )pjmin. Lag.

L2n+1 = πi (L̃)
cplx. Leg.

HP ntot. cplx.

S3
N 2n

Ñ n

π

We use the standard Riemannian ﬁbration πi . Then L̃2n+1 → S 4n+3 (1)is a minimal Legendrian immersion relative to I.
Therefore, L̃2n+1 → Ci P 2n+1 is a minimal Lagrangian immersion relative
to J, which has the S 1 -action induced by the orthogonal complex
structure I.

S

Hn+1

S 4n+3 (1)πi

Ci P 2n+1
S3
π
HP n

min. Leg.
L̃2n+1 = πj−1 (Ñ )min. Lag.
covering

L2n+1 = πi (L̃)

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Theorem (Tsukada, 1985)
Any n-dimensional totally complex submanifold Ñ in HP n with parallel
second fundamental form is locally congruent to one of the following
immersed totally complex submanifolds:(0) CP 1 −→ RP 2 ⊂ S 4 = HP 1
(1)

CP n ⊂ HP n

Veronese min surf.

totally geodesic(2) Sp(3)/U (3) −→ HP 6(3) SU (6)/S(U (3) × U (3)) −→ HP 9
(4)

SO(12)/U (6) −→ HP 15(5)

E7 /((U (1) × E6 )/Z3 ) −→ HP 27(6)

CP 1 (c̃) × CP 1 (c̃/2) −→ HP 2

(7)

CP 1 (c̃) × CP 1 (c̃) × CP 1 (c̃) −→ HP 3

(8)

CP 1 (c̃) ×SO(n + 1)−→ HP nSO(2) × SO(n − 1)(n ≥ 4)

All of those totally complex submanifolds are obtained from the follwing
compact homogeneous complex Legendrian submanifolds embedded in
CP 2n+1 .
(1) CP n ⊂ CP 2n+1
(2) Sp(3)/U (3) −→ CP 13(3) SU (6)/S(U (3) × U (3)) −→ CP 19(4) SO(12)/U (6) −→ CP 31(5) E7 /(U (1) × E6 )/Z3 −→ CP 55(6) CP 1 (c̃) × CP 1 (c̃/2) −→ CP 5
(7) CP 1 (c̃) × CP 1 (c̃) × CP 1 (c̃) −→ CP 7(8) CP 1 (c̃) ×SO(n + 1)= CP 1 (c̃) × Qn−1 (C) −→ CP 2n+1
SO(2) × SO(n − 1)(n ≥ 4)

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Quaternionic Symmetric space
A Riemannian manifold M is called a quaternionic symmetric space if M
satisﬁes the following conditions:(i) M is a quaternionic Kähler manifold with quaternionic structure Q.
(ii) M is a symmetric space.
(iii) Qp is contained in the linear holonomy group for some point p in M .
G
Sp(n + 1)
SU (n + 3)
SO(n + 5)
G2
F4
E6
E7
E8

K
Sp(n) × Sp(1)S(U ((n + 1) × U (2))
SO(n + 1) × SO(4)(Sp(1) × Sp(1))/Z2(Sp(3) × Sp(1))/Z2(SU (6) × SU (2))/Z2(Spin(12) × SU (2))/Z2(E7 × SU (2))/Z2

dim M = G/K
4n
4n
4n
8
28
40
64
112

TheoremSuppose that N 2n ⊂ HP n is a totally complex submanifold with∇∗ αN = 0.
b 2n+3 = π −1 (N ) ⊂ S 4n+3 is an “R-space”
Then its inverse image Nbassociated to a quaternionic symmetric space G/K with ∇∗ αN ̸= 0.
b = π −1 (N ) ⊂ S 4n+3 is a singular orbit of a quaternionicNsymmetric space G/K.
πi
S 4n+3 (1)Ci P 2n+1
pi
S3
πj
π
b =π
N

−1

(N )

Cj P 2n+1

pj

HP n

S3
π

tot. cplx.
N 2nSubmanifolds of Symmetric Spaces and Their Time Evolutions

Deﬁnition
Let (M, g) be a Riemannian manifold and ∇ Levi-Civita connection of g.
A linear connection ∇c on M is called a canonical connection if itsatisﬁes:(1) ∇c g = 0,(2) ∇c D = 0,
where D := ∇ − ∇c .

Theorem (Olmos-Sánchez, 1991)
Let M be a connected compact Riemannian submanifold of Rn and let α
be its second fundamental form. Then the following three statements areequivalent:(1) M admits a canonical connection ∇c satisfyin ∇c α = 0.
(2) M is a homogeneous submanifold with constant principal curvatures.
(3) M is a standard embedding of an R-space.

b = π −1 (N ), there exist ∇c := ∇ − D such thatOn Nb∇c αN = 0,∇c D = 0.

b is an R-space associatedFrom Olmos-Sánchez’s result we obtain that N
to a quaternionic symmetric space G/K.
b.
We can explicitly construct a tensor ﬁeld D of type (1, 2) on Nn
Suppose that N is a totally complex submanifold of HP .
b , I N , J N , K N ∈ Cj P 2n+1 satisﬁes
For each x ∈ N
π(x) π(x)
π(x)
N
Iπ(x)
(Tπ(x) N ) ⊥ Tπ(x) N,NJπ(x)(Tπ(x) N ) = Tπ(x) N,N
Kπ(x)⊥ Tπ(x) N.

There are λ1 (x), λ2 (x), λ3 (x) ∈ Sp(1) such thatNIπ(x)(x) = xλ1 (x),NJπ(x)(x) = xλ2 (x),NKπ(x)(x) = xλ3 (x).

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We deﬁne a tensor ﬁeld D of type (1, 2) as follow:
b , for each X, Y ∈ Tπ(x) N , V ∈ Vx N
b , v1 , v2 , v3 ∈ R,
At x ∈ N
N
N
N
N• DX̃ (Ỹ ) = ⟨(J^
π(x) X), Ỹ ⟩Jπ(x) (x) = gN (Jπ(x) X, Y )Jπ(x) (x) ∈ Vx N̂ ,N
• DX̃ (V ) = v2 X̃λ2 (x) = −v2 (J^π(x) X) ∈ Hx N̂

(∀ V = x3X!

va λa (x) ),

a=1

!
3
X
v2
v2 ^
N X) ∈ H N̂• DV (X̃) = X̃λ2 (x) = − (Jπ(x)va λa (x) ),(∀ V = x
x
2
2
a=1

1
• DU (V ) = x (v2 u3 + v3 u2 )λ1 (x) + (v3 u1 − v1 u3 )λ2 (x)21+ (− v1 u2 − v2 u1 )λ3 (x)
2
v1

v2
1
= x u1− u2
2
λ1 (x) λ2 (x)v3u3∈ Vx N̂ .

λ3 (x)

Here note that
NN
(J^
π(x) X) = (Jπ(x) X)(x) = −(X̃)x λ2 (x).
xbFor each w1 , w2 , w3 ∈ T N

Lemma
gNb (Dw2 , w3 ) + gNb (w2 , Dw1 w3 ) = 0b∇N D = D · D

i.e. ∇c g = 0i.e. ∇c D = 0

Lemma
Suppose that N is a 2n-dimensional totally complex submanifold of HP nwith parallel second fundamental form. Then the second fundamentalbb satisﬁes the equation
form αN of Nbbb∇∗w1 αN (w2 , w3 ) + αN (Dw1 w2 , w3 ) + αN (w2 , Dw1 w3 ) = 0.
i.e.

b∇c α N = 0

Submanifolds of Symmetric Spaces and Their Time Evolutions

totally cplx. imm. Ñ

q. symm. sp. G/K
G2(Sp(1) × Sp(1))/Z2
SU (n + 1)
S(U (2) × U (n))
F4(Sp(3) × Sp(1))/Z2
E6(SU (6) × Sp(1))/Z2

CP 1
CP n
Sp(3)
U (3)
SU (6)S(U (3) × U (3))
SO(12)U (6)E7(U (1) × E6 )/Z3
CP 1 (c̃) × CP 1 (c̃/2)
CP 1 (c̃) × CP 1 (c̃) × CP 1 (c̃)
CP 1 (c̃) ×75

SO(n + 1)
SO(2) × SO(n − 1)E7(Spin(12) × SU (2))/Z2
E8(E7 × SU (2))/Z2
SO(7)
SO(4) × SO(3)SO(8)SO(4) × SO(4)SO(n + 5)
SO(4) × SO(n + 1)

(n = 2)(n = 3)(n ≥ 4)(G, K) = (G2 , SO(4))
In this case we obtain the Chiang Lag submfd as L3 = πi (πj−1 (Ñ )).

∼=g2 = so(4) ⊕ p

S

H2

πi

S 7 (1)πj

πCj P 3

2

(RP )

pj

S 4 (1) = HP 1

S

π

pi

S3

SO(4)/O(2)−1

Ci P 3L̃3 = πj−1 (CP 1 )

Z4

L3 = πi (L̃)
cplx. Leg.

tot. cplx.

S3
πÑ 2 = CP 1

Z2

RP 2 = N 2

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References I[1] R. Chiang, New Lagrangian submanifolds of CP n . Int. Math. Res.
Not. 45, 2437–2441 (2004)[2] B. Y. Chen, F. Dillen, L. Verstraelen, and L. Vrancken, An exotic
totally real minimal immersions of S 3 in CP 3 and its characterization,
Proc. Royal Soc. Edinburgh Sect. A, Math. 126 (1996), 153–165.
[3] L. Bedulli and A. Gori, A Hamiltonian stable minimal Lagrangiansubmanifold of projective space with nonparallel second fundamental
form. Transform. Groups 12 (2007), no. 4, 611–617.
[4] J. D. Evans and Y. Lekili, Floer cohomology of the Chiang
Lagrangian, Selecta Math. (N.S.) 21 (2015), no. 4, 1361–1404.
[5] Y. Ohnita, Stability and rigidity of special Lagrangian cones overcertain minimal Legendrian orbits, Osaka J. Math. 44 no. 2 (2007),305-334.

References II[6] Y. Ohnita, On deformation of 3-dimensional certain minimal
Legendrian submanifolds, Proc. The 13-th International Workshop
on Diﬀerential Geometry and Related Fields, 13 (2009), pp.71–87,
NIMS, KMS and Grassmann Research Group.
[7] K.Tsukada, Parallel submanifolds in a quaternion projective space.
Osaka J. Math. 22 (1985), 187–241.
[8] C. Olmos and C. Sánchez, A geometric characterization of the orbitsof s-representations. J. reine angew. Math. 420 (1991), 195–202.
[9] Y. Ohnita, Parallel Kähler submanifolds and R-spaces. (submitted),a preprint, OCAMI Preprint Ser. 20-21.
[10] J.-T. Cho, K. Hashimoto and Y. Ohnita, Totally complex
submanifolds and R-spaces. in preparation.

Submanifolds of Symmetric Spaces and Their Time Evolutions

Thank you very much for your kind attention!!

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Minimal PF submanifolds in Hilbert spaces
with symmetries
Masahiro Morimoto
There are several kinds of minimal submanifolds which have certain symmetries. A
submanifold M of a Riemannian manifold N is called austere ([1]) if for each normal
vector ξ the eigenvalues with multiplicities of the shape operator Aξ is invariant underthe multiplication by (−1). M is called reflective ([3]) if it is a connected component of
the fixed point set of an involutive isometry of N . M is called weakly reflective ([2]) if
for each normal vector ξ at each p ∈ M there exists an isometry νξ of N satisfying theconditions νξ (p) = p, dνξ (ξ) = −ξ and νξ (M ) = M . From these definitions we havereflective⇒

weakly reflective⇒

austere⇒minimal.

It is an interesting problem to classify or give examples of these minimal submanifolds.
A fundamental class of submanifolds in Hilbert spaces is given by proper Fredholm(PF) submanifolds in Hilbert spaces ([6]). Roughly speaking they are submanifolds
in Hilbert spaces where the shape operators are compact operators and the distancefunctions are compatible with the Palais-Smale condition. Many examples of PF submanifolds are obtained through a Riemannian submersion ΦK : Vg → G/K which iscalled the parallel transport map ([7]). Here G/K is a compact normal homogeneous
space and Vg := L2 ([0, 1], g) the Hilbert space of all L2 -paths with values in the Liealgebra g of G. It is known that if M is a compact submanifold of G/K then the
inverse image Φ−1
K (M ) is a PF submanifold of Vg .
In my talk I will introduced the concept of reflective submanifolds, weakly reflective
submanifolds and austere submanifolds into the class of PF submanifolds in Hilbert
spaces and show that under suitable condition if M is a weakly reflective submanifoldof G/K then the inverse image Φ−1
K (M ) is a weakly reflective PF submanifold of Vg .
References[1] R. Harvey and H. B. Lawson, Jr., Calibrated geometries, Acta Math., 148 (1982), 47-157.
[2] O. Ikawa, T. Sakai, H. Tasaki, Weakly reflective submanifolds and austere submanifolds. J. Math.
Soc. Japan 61 (2009), no. 2, 437-481.
[3] Dominic S. P. Leung, The reflection principle for minimal submanifolds of Riemannian symmetric spaces, J. Differential Geom. 8 (1973), 153-160.
[4] M. Morimoto, On weakly reflective PF submanifolds in Hilbert spaces, to appear in Tokyo J.
Math.
[5] M. Morimoto, Austere and arid properties for PF submanifolds in Hilbert spaces, Differential
Geom. Appl., Vol. 69 (2020) 101613.
[6] C.-L. Terng, Proper Fredholm submanifolds of Hilbert space. J. Differential Geom. 29 (1989), no.
1, 9-47.
[7] C.-L. Terng, G. Thorbergsson, Submanifold geometry in symmetric spaces. J. Differential Geom.
42 (1995), no. 3, 665-718.

Submanifolds of Symmetric Spaces and Their Time Evolutions

79(M. Morimoto) Osaka City University Advanced Mathematical Institute, 3-3-138,
Sugimoto, Sumiyoshi-ku, Osaka, 558-8585, Japan
E-mail address: mmasahiro0408@gmail.com

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Sec. 4

Sec. 5

Sec. 6

Minimal PF submanifolds in Hilbert spaces
with symmetries
Masahiro Morimoto
Osaka City University Advanced Mathematical InstituteSubmanifolds of Symmetric Space and Their Time Evolution
March 5 - 6, 2021

Sec. 1

Sec. 2

Sec. 3

Sec. 4

Sec. 5

Sec. 6

Overview
M

En
Euclidean sp.

Msubmanifold

,→

⇝

f

submanifold

,→

N

M

Riemannian mfd(ﬁnite dim)

submanifoldminimal submanifold with symmetry• reﬂective submanifold• weakly reﬂective submanifold −→• austere submanifold,→

V
Hilbert sp.
(inﬁnite dim)we deﬁne and study• reﬂective submanifold• weakly reﬂective submanifold• austere submanifold

Submanifolds of Symmetric Spaces and Their Time Evolutions
Sec. 1

Sec. 2

Sec. 3

Sec. 481
Sec. 5

Sec. 6Plan

1Minimal submanifolds with symmetries

2

Submanifolds in Hilbert spaces

3

The parallel transport map

4

Submanifold geometries via the parallel transport map

5

Minimal PF submanifolds with symmetries

6

Symmetric properties
via the parallel transport map (: Main results)

Sec. 1

Sec. 2

Sec. 3

Sec. 4

Sec. 5

Sec. 6

Sec. 1 - Minimal submanifolds with symmetries (1/4)
Deﬁnition (Harvey-Lawson 1982)
Let

M

,→

submanifold

N
Riem. manifold.

M is austere⇔ ∀p ∈ M , ∀ξ ∈ T p⊥ M , the eigenvalues (with multiplicities)

defof the shape operator AξM is invariant under the multiplication by (−1).

Proposition
austere=⇒minimalProblemClassify (or, ﬁnd examples of) austere submanifolds.

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Sec. 1 - Minimal submanifolds with symmetries (2/4)

Theorem (F. Podestà 1997)
G: Lie group, N : Riemannian manifold.
Suppose G ↷ N : isometric action of cohomogeneity one.
(i.e. codim (principal G-orbit) = 1).
⇒ any singular (i.e. non-principal) G-orbit is an austere submanifold of N .
NoteMore precisely, Podestà proved that any singular G-orbit M satisﬁes:∀p ∈ M , ∀ξ ∈ T p⊥ M , there exists νξ ∈ Isom(N) satisfying

Sec. 1νξ (p) = p,νξ (M) = M,

Sec. 2

Sec. 3dνξ (ξ) = −ξ.

Sec. 4

Sec. 5

Sec. 1 - Minimal submanifolds with symmetries (3/4)
Deﬁnition (Ikawa-Sakai-Tasaki 2009)
Let

,→

M
submanifold

N
Riem. manifold.

M is weakly reﬂective⇔ ∀p ∈ M , ∀ξ ∈ T p⊥ M , there exists νξ ∈ Isom(N) satisfyingdefνξ (p) = p,νξ (M) = M,

Proposition (Ikawa-Sakai-Tasaki 2009)
reﬂective ⇒ weakly reﬂective ⇒dνξ (ξ) = −ξ.

austere⇒minimal(D.S. Leung 1973)

ProblemClassify (or, ﬁnd examples of) weakly reﬂective submanifolds.

Sec. 6

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Sec. 1 - Minimal submanifolds with symmetries (4/4)

Study on weakly reﬂective submanifolds
Ikawa-Sakai-Tasaki (2009) classiﬁed weakly reﬂective submanifolds
and austere submanifolds which are orbits of s-representations.
Ohno (2016) gave examples of weakly reﬂective submanifolds which
are orbits of Hermann actions K2 ↷ G/K1 .
Enoyoshi (2018) showed that there exists unique weakly reﬂective
f 3 (Im O)
principal orbits in the cohomogeneity one action G2 ↷ GrNote
All known examples of weakly reﬂective submanifolds are homogeneous.

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Sec. 2 - Submanifolds in Hilbert spaces (1/6)
Basic Setting
Suppose: ∀p ∈ M , T p M is a closed subsp. of V . (⇝ T ⊥ M )smooth immersion

M
Hilbert manifoldg∇Levi-Civita conn.

V

separable Hilbert sp.

fh·, ·iinner product⇝⇝Riem. metric,→D
Levi-Civita conn.

second fundm. form α，shape op. A，normal conn.∇⊥ is deﬁned:(
Gauss formula.
DX Y = ∇X Y + α(X, Y), X, Y ∈ Γ(T M).
 Weingarten formula DX ξ = −Aξ X + ∇⊥
ξ ∈ Γ(T ⊥ M).
X ξ,
Difﬁculty: Spectral theory of shape op. Aξ : T p M → T p M .

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Sec. 2 - Submanifolds in Hilbert spaces (2/6)
Suppose (C.-L. Terng 1989)
M has ﬁnite codimension in V .
The end point map Y : T ⊥ M → V , (p, ξ) 7→ p + ξsatisﬁes: ∀r > 0, restriction to Dr (: normal disc bdl of radius r)

Y|Dr : Dr → V

is proper and Fredholm (i.e. differential is a Fredholm op.)．
Then M is called a proper Fredholm submanifold (PF submanifold).

Sec. 1

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Sec. 2 - Submanifolds in Hilbert spaces (3/6)

ExampleIf V = En thenM ,→ En is PF ⇔ immersion M ,→ En is proper.
If dim V = ∞, then the unit sphere if V is not PF.
Every afﬁne subspace of a Hilbert space V is PF.
Every orbit of the gauge transformations is PF. (page 6/6)

Sec. 6

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Sec. 2 - Submanifolds in Hilbert spaces (4/6)
Proposition (C.-L. Terng 1989)
If M is a PF submanifold of V , then
(1) the shape operator Aξ : T p M → T p M is a self-adjoint compact op.
(2) for each u ∈ V , the function fu : M → R, p 7→ kp − uk2satisﬁes Condition C (Palais-Smale 1960s).

RemarkAξ : T p M → T p M real eigenvaluesµ1 < µ 2 < · · · < 0 < · · · < λ 2 < λ 1| {z }| {z }

ﬁnite multip.
ﬁnite multip
These are called the principal curvatures of M in direction ξ .

the shape op. Aξ : T p M → T p M is not of trace class in general(There is no natural deﬁnition for the mean curvature.)

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Sec. 2 - Submanifolds in Hilbert spaces (5/6)
Deﬁnition (King-Terng 1993, Heintze-Liu-Olmos 2006, Koike 2002)
M : PF submanifold，ξ ∈ T ⊥ M ，Aξ : T p M → T p M : shape operator.
 µ1 ≤ µ2 ≤ · · · < 0 < · · · ≤ λ2 ≤ λ1 : eigenvalues of Aξ {κk }∞
k=1 : distinct eigenvalues of Aξ s.t. |κk |>|κk+1 | or κk = −κk+1 ．
PP∞ss(1) ζ -regularized mean curvature trζ Aξ := lim s&1 ( ∞k=1 λk − k=1 |µi | )P(2) regularized mean curvature trr Aξ := ∞k=1 (λk + µk )P∞(3) formal mean curvature
tr f Aξ := k=1 multip(κk )κk(1) M is ζ -minimal ⇔ ∀ξ ∈ T ⊥ M , trζ Aξ = 0
def(2) M is r-minimal ⇔ ∀ξ ∈ T ⊥ M , trr Aξ = 0 (and tr A2 < ∞)def(3) M is f -minimal ⇔ ∀ξ ∈ T ⊥ M , tr f Aξ = 0
def

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Sec. 2 - Submanifolds in Hilbert spaces (6/6)
Example (Terng 1989, Pinkall-Thorbergsson 1990, Terng 1995): orbits of the gauge transformations．

G: conn. compact Lie group with bi-inv. Riem met., g: its Lie alg.
P → [0, 1]: trivial principal G-bundle ( i.e. P := [0, 1] × G)．
(transitive)(gauge transf. gp) gauge(connections of P)
transf.

G := H 1 ([0, 1], G)∪V := L2 ([0, 1], g)

0 g−1

g ∗ u:=gug − g g(g ∈ G, u ∈ Vg )↷{g ∈ G | (g(0), g(1)) ∈ H}(∀H : closed subgp of G × G)

Sec. 1

Sec. 2path sp.
(Hilbert sp.)Vg := L2 ([0, 1], g)∪P(G, H)-orbit↑=:

subgroup P(G, H)=

path group(Hilbert Lie group)↷−1

PF submanifold !

Sec. 3

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Sec. 6

Sec. 3 - The parallel transport map (1/5)
Setting
G: conn. compact Lie group with bi-invariant Riem met., g: its Lie algebra.
G := H 1 ([0, 1], G) : path group of all Sobolev H 1 -paths from [0, 1] to G.
Vg := L2 ([0, 1], g): path space of all L2 -paths from [0, 1] to g.
Deﬁnition (Terng 1995, Terng-Thorbergsson 1995)
Φ : Vg →G∈∈

The parallel transport map is deﬁned bydef

7→ Φ(u) := gu (1).
( −1 0
gu gu = u,Here, gu ∈ G is deﬁned by the ODE
gu (0) = e ∈ G.
u

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Sec. 3 - The parallel transport map (2/5)
Deﬁnition
Deﬁne the map Ψ : G → G × G by Ψ(g) := (g(0), g(1)) for g ∈ G.

Proposition (Terng 1995)
The parallel transport map Φ : Vg → G is equivariant with respect to Ψ.
That is, the following diagram commutes:
gauge transformationg ∗ u := gug−1 − g0 g−1
G
↷
Vg
Ψ↓

Φ↓

G
G×G ↷
(b1 , b2 ) · a := b1 ab−12isometric action
That is, Φ(g ∗ u) = Ψ(g) · Φ(u).

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Sec. 3 - The parallel transport map (3/5)
Theorem (Terng-Thorbergsson 1995)(1) Φ is a Riemannian submersion，(2) any two ﬁbers of Φ are congruent
under the isometry of Vg ，(3) Φ is a principal P(G, {e}×{e})-bdl，(4) If M is a closed submfd of G，then Φ−1 (M) is a PF submfd of Vg ．
(5) If M = H · a for subgp H ⊂ G × G
then Φ−1 (M) = P(G, H)∗u foru ∈ Φ−1 (a).

Nice points(1) We can obtain examples of (homogeneous) PF submanifolds.
(2) We can linearize geometrical problems of submanifold M ⊂ G.

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Generalization (Terng-Thorbergsson1995)
G/K : compact normal homg. sp，π : G → G/K : projection．
The parallel transport map over G/K
def

ΦG/K := π ◦ Φ : Vg → G → G/K .
(1) ΦG/K : a Riemannian submersion，(2) Two ﬁbers of ΦG/K are congruent
under the isometry of Vg ，(3) ΦG/K : a principal P(G, {e}× K)-bdl.
(4) If M is a closed submfd of G/K ，−1 (M) is a PF submfd of V ．
then ΦG/Kg(5) If M = K 0 · aK for subgp K 0 ⊂ G−1 (M) = P(G, K 0 × K)∗u forthen ΦG/K−1 (aK).
u ∈ ΦG/K

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Sec. 3 - The parallel transport map (5/5)
Fundamental problem−1 (M) ?
: The geometrical relation between M and ΦG/K

E.g.

−1 (M) is ζ -minimal and r -minimal.
• M : minimal ⇒ ΦG/K

(King-Terng 1993, Heintze-Liu-Olmos 2006)．
• Suppose G/K : symmetric space of compact type.
−1 (M):isoparametric (Terng-Thorbergsson1995).
M : equifocal ⇒ ΦG/K

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Sec. 4 - Submanifold geometries via the p.t.m. (1/5)
Problem
G/K ：cpt. normal homog. sp.，g = k + n：orthogonal decomp.
• M ：closed submanifold of G/K through eK ∈ M .
⊥ M  T ⊥ Φ−1 (M)．
• Fix a normal vector ξ ∈ T eK
G/K0̂

 Φ−1 (M): second f. form
α G/K



−1 (M)
 ΦG/K⇝: shape op.
Aξ





{µ}
: principal curv.
 M
α: second f. form



 M
Aξ : shape op.
⇝ {λ} : principal curv.
The relation ?

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Sec. 4 - Submanifold geometries via the p.t.m. (2/5)
Theorem (M. 2019)：the second fundamental form
−1 (M),∀X, Y ∈ T 0̂ ΦG/K

−1 (M)ΦG/K

α(X, Y) = α M

R1

R1



X(t)n dt, 0 Y(t)n dt
⊥
R
⊥
R
R1
1
1
1
+ 12 0 X(t)k dt, 0 Y(t)n dt − 12 0 X(t)n dt, 0 Y(t)n dt
R⊥  R h R
i ⊥
R1
1
1
t
+ 21 0 X(t)dt, 0 Y(t)dt − 0 0 X(s)ds, Y(t) dt ．
R

0Theorem (M. 2019)：the shape operator−1 (M), ξ ∈ T ⊥ Φ−1 (M),∀X ∈ T 0̂ ΦG/K
0̂ G/K
R

R

R
>
−1 (M)
ΦG/K
1
1
1
1
1
M
Aξ
(X) = Aξ 0 X(t)n dt − 2 0 X(t)n dt, ξ + 2 0 X(t)k dt, ξkR> h R⊥i R 1 R t
1t
1− 2 0 X(t)dt, ξ + 0 X(s)ds, ξ − 0 0 X(s)dsdt, ξ ．

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Theorem (Koike 2002, M. 2019)：Principal curvatures
G/K ：compact symmetric sp．g = k + n: canonical decomp．
Suppose M ：curvature adapted submfd of G/K ．
(i.e. ad(ξ)2 : n → n preserves T eK M
and commutes with AξM : T eK M → T eK M .)√{λ}: eigenvalue of AξM ，{ −1ν}: eigenvalue of ad(ξ) : g → g.
−1 (M) in direction ξ isThen the principal curvatures of ΦG/K
)
(
ν
ν
0, λ,
,
λ, ν > 0, n ∈ Z\{0}, m ∈ Z.
nπ arctan λν + mπ
eigenfunctions and multiplicities are given in the next page:Sec. 1

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Sec. 4 - Submanifold geometries via the p.t.m. (4/5)

Theorem (Koike 2002, M. 2019)：Principal curvatures
Setµ(ν, λ, m) :=νarctan λν + mπ．

eigenval.

basis of eigenfunctions

multip.

0{xi0 sin nπt, y(0,λ)
cos nπt, y(0,⊥)
cos nπt}n∈Z≥1 , λ, i, j, l
j
l

∞

λ
ν
nπ
µ(ν, λ, m){y(0,λ)}jjm(0, λ)nP{xr(ν,⊥) sin nπt − y(ν,⊥)
cos nπt}r
r

o

(ν,λ)
ν
sin nπt + y(ν,λ)
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By using the formula of the shape operator, we obtain:

Theorem (M. 2019): The totally geodesic property
The following are equivalent:−1 (M) is a totally geodesic PF submanifold of V
(1) ΦG/K
g(i.e. an afﬁne subspace of Vg ).
⊥ M ⊂ (center of g).
(2) M is a totally geodesic submfd of G/K s.t. T eK
−1 (M) is not totally geodesic, except for rare cases
Thus ΦG/K

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Sec. 5 - Minimal PF submanifolds with symmetries (1/4)
Deﬁnition
Let

M
PF submfd,→

V
separable Hilbert sp.

M is reﬂective⇔ ∃ σ : V → V : involutive isometry s.t. M = Fix(σ)0 .
def
M is weakly reﬂective⇔ ∀p ∈ M , ∀ξ ∈ T p⊥ M , ∃νξ : V → V : isometrydefs.t. (i) νξ (p) = p, (ii) dνξ (ξ) = −ξ , (iii) νξ (M) = M .
M is austere⇔ ∀p ∈ M , ∀ξ ∈ T p⊥ M ,def{eigenvalues of Aξ (with multip)} is invariant under (−1)×.

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Sec. 5 - Minimal PF submanifolds with symmetries (2/4)

Recall,→Msubmfdreﬂective⇒

N
Riem mfd⇒

weakly reﬂective

austere⇒minimalNote−1 (M) ,→ΦG/KPF submfd

Vg

separable Hilbert sp.

reﬂective ⇒ weakly reﬂective ⇒ austere ⇒Sec. 1

Sec. 2

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ζ -minimalr-minimal

Sec. 5)
same

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Sec. 5 - Minimal PF submanifolds with symmetries (3/4)
Theorem (M. 2019)
G/K : compact normal homogeneous space.
Then each ﬁber of the parallel transport map ΦG/K : Vg → G/K
is a weakly reﬂective PF submanifold of Vg .

Proof (outline)
The canonical reﬂection (M. 2019) of the path space Vg = L2 ([0, 1], g)r : Vg → Vg , u 7→ r(u) , r(u)(t) := −u(1 − t)plays an important role.

RemarkThe ﬁber is not totally geodesic (except for rare cases).
Therefore, the ﬁber is not reﬂective.

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Problem−1 (M) : reﬂective ?
(1) M : reﬂective⇒ ΦG/K−1 (M) : weakly reﬂective ?
(2) M : weakly reﬂective ⇒ ΦG/K
−1 (M) : austere ?
(3) M : austere⇒ ΦG/K

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Sec. 6 - Symmetric properties via the p.t.m. (1/7)
Problem−1 (M) : reﬂective × (e.g. ﬁber)(1) M : reﬂective⇒ ΦG/K−1 (M) : weakly reﬂective(2) M : weakly reﬂective ⇒ ΦG/K
(Thm A and B)−1(3) M : austere⇒ ΦG/K (M) : austere △ (Thm C and D)

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Theorem A (M. 2019)
G: connected compact semi-simple Lie groupwith bi-inv metric induced from negative of the Killing form,
K : symmetric subgroup of G s.t. (G, K): effective.
M : weakly reﬂective submanifold of G/K .
−1 (M) is a weakly reﬂective PF submanifold of V .
⇒ ΦG/Kg

Theorem B (M. 2020)
N : compact isotropy irreducible Riemannian homogeneous space.
Fix p ∈ N . Set G := Isom0 (N) and K := Isom(N) p so that N  G/K .
M : weakly reﬂective submanifold of N  G/K−1 (M) is a weakly reﬂective PF submanifold of V .
⇒ ΦG/Kg

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Sec. 6 - Symmetric properties via the p.t.m. (3/7)

Theorem C (M. 2020)
Suppose G/K = S O(n + 1)/S O(n): sphere.
M : closed submanifold of G/K .
Then the following conditions are equivalent:(1) M is an austere submanifold of G/K ,−1 (M) is an austere PF submanifold of V .
(2) ΦG/Kg

Theorem D (M. 2021)
Let G/K : symmetric of compact type. Suppose G is simple.
M : an austere orbit of a Hermann action with commuting involutions.
−1 (M) is an austere PF submanifold of V .
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ExampleG/K : symmetric sp. of compact type, K 0 : closed subgroup of G.
Suppose the action K 0 ↷ G/K : of cohomogeneity one.
Let M := K 0 · aK : singular orbit through aK ∈ G/K .
(⇒ M is weakly reﬂective submfd of G/K (Podesta 1997, IST2009).
−1 (M) (= P(G, K 0 × K)∗u) is a weakly reﬂective PF submfd
By Thm A, ΦG/K
of Vg .
Example(U, L): compact Riem. symmetric pair. Assume L: connected.
u = l + p: canonical decomp. Ad : L → S O(p) : isotropy rep.
Suppose M := Ad(L)x: weakly reﬂective submfd of S (kxk)(⊂ p).
(Such M was classiﬁed in Ikawa-Sakai-Tasaki 2009)
−1 (M) = P(S O(p), Ad(L) × S O(p) ) ∗ 0̂ is a
By Thm A (or Thm B), ΦG/Kx
weakly reﬂective PF submanifold of Vg .

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Sec. 6 - Symmetric properties via the p.t.m. (5/7)
ExampleG/K : symmetric space of compact type. K 0 : symmetric subgroup of G.
Suppose M = K 0 · aK : weakly reﬂective orbit of the Hermann action.
(Such examples were given by Ohno 2016).
−1 (M) (= P(G, K 0 × K)∗u) is a weakly reﬂective PF
Then by Thm A, ΦG/K
submfdExampleG2 ：exceptional Lie grp，G̃r3 (Im O) ( S O(7)/(S O(3) × S O(4)))：Grassmann manifold
G2 ↷ G̃r3 (Im O) has unique weakly reﬂective orbit M [Enoyoshi 2018].
−1
Then by Thm A, ΦG/K (M) = P(S O(7), G2 × S O(3) × S O(4)) ∗ 0̂ is aweakly reﬂective PF submfd of Vo(7) ．

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Example(U, L): compact Riem. symmetric pair. Assume L: connected.
u = l + p: canonical decomp. Ad : L → S O(p) : isotropy rep.
Suppose M := Ad(L)x: austere submfd of S (kxk)(⊂ p).
(Such M was classiﬁedby Ikawa-Sakai-Tasaki 2009)−1Then by Thm C , ΦG/K (M) = P(S O(p), Ad(L) × S O(p) x ) ∗ 0̂ is an austere
PF submanifold of Vg .
ExampleG/K : symmetric space of compact type. Suppose G is simple.
Suppose M : austere orbit of a Hermann action with commuting involutions.
(Such examples classiﬁed by Ikawa 2011).
−1 (M) (= P(G, K 0 × K)∗u) is an austere PF submfd ofThen by Thm D, ΦG/K
Vg .

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Sec. 6 - Symmetric properties via the p.t.m. (7/7)

Note
Recently, Taketomi (2018) introduced a generalized concept of weakly
reﬂective submanifolds, namely arid submanifolds:⇒ austere ⇒ minimal⇒arid

reﬂective ⇒ weakly reﬂective ⇒A submanifold M of a Riemannian manifold N is arid
⇔ ∀p ∈ M , ∀ξ ∈ T p⊥ M there exists ϕξ ∈ Isom(N) satisfyingdefϕξ (p) = p,ϕξ (M) = M,

dϕξ (ξ) , ξ.

Theorems A and B can be formulated to the arid case.

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97
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On homogeneous minimal submanifolds

Note
In inﬁnite dimensional Hilbert spaces,
there exist many examples of homogeneous minimal PF submanifolds
which are not totally geodesic (by Theorem in Sec. 4.)

Theorem (Di Scala 2002)
In ﬁnite dimensional Euclidean spaces,
any homogeneous minimal submanifolds must be totally geodesic.

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Reference I[1]

J. Berndt, L. Vanhecke, Curvature-adapted submanifolds, Nihonkai
Math. J. 3 (1992), no. 2, 177–185.

[2]

K. Enoyoshi, Principal curvatures of homogeneous hypersurfaces in af3 (Im O) by the G2 -action, to appear in Tokyo
Grassmann manifold Gr
J. Math.

[3]

R. Harvey and H. B. Lawson, Jr., Calibrated geometries, Acta Math.,
148 (1982), 47-157.

[4]

E. Heintze, X. Liu, C. Olmos, Isoparametric submanifolds and a
Chevalley-type restriction theorem, Integrable systems, geometry,
and topology, 151-190, AMS/IP Stud. Adv. Math., 36, Amer. Math.
Soc., Providence, RI, 2006.

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Reference II[5]

E. Heintze, R. Palais, C.-L. Terng, G. Thorbergsson, Hyperpolar
actions on symmetric spaces, Geometry, topology, & physics,
214-245, Conf. Proc. Lecture Notes Geom. Topology, IV, Int. Press,
Cambridge, MA, 1995.

[6]

O. Ikawa, T. Sakai, H. Tasaki, Weakly reﬂective submanifolds andaustere submanifolds. J. Math. Soc. Japan 61 (2009), no. 2, 437-481.

[7]

O. Ikawa, The geometry of symmetric triad and orbit spaces ofHermann actions, J. Math. Soc. Japan 63 (2011), no. 1, 79-136.

[8]

C. King, C.-L. Terng, Minimal submanifolds in path space, Global
analysis in modern mathematics, 253-281, Publish or Perish,
Houston, TX, 1993.

[9]

N. Koike, On proper Fredholm submanifolds in a Hilbert space arisingfrom submanifolds in a symmetric space, Japan. J. Math. (N.S.) 28(2002), no. 1, 61-80.

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Reference III[10] M. Morimoto, On weakly reﬂective PF submanifolds in Hilbert spaces,
to appear in Tokyo J. Math.
[11] M. Morimoto, Austere and arid properties for PF submanifolds in
Hilbert spaces, Differential Geom. Appl., Vol. 69 (2020) 101613.
[12] M. Morimoto, On weakly reﬂective submanifolds in compact isotropy
irreducible Riemannian homogeneous spaces, arXiv:2003.04674.
[13] M. Morimoto, Minimal PF submanifolds in Hilbert spaces withsymmetries, Ph.D. Thesis. doi/10.24544/ocu.20200615-005[14] M. Morimoto, Curvatures and austere property of orbits of path group
actions induced by Hermann actions, in preparation.
[15] S. Ohno, A sufﬁcient condition for orbits of Hermann actions to be
weakly reﬂective, Tokyo J. Math. 39 (2016), no. 2, 537-564.

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Reference IV[16] R. S. Palais, Morse theory on Hilbert manifolds, Topology 2 (1963),
299-340.
[17] R. S. Palais, C.-L. Terng, Critical Point Theory and Submanifold
Geometry, Lecture Notes in Math., vol 1353, Springer-Verlag, Berlinand New York, 1988.
[18] U. Pinkall, G. Thorbergsson, Examples of inﬁnite dimensional
isoparametric submanifolds, Math. Z. 205 (1990), no. 2, 279-286.
[19] F. Podestà, Some remarks on austere submanifolds, Boll. Un. Mat.
Ital. B (7) 11 (1997), no. 2, suppl., 157-160.
[20] A. J. D. Scala, Minimal homogeneous submanifolds in Euclidean
spaces, Ann. Global Anal. Geom. 21 (2002), no. 1, 15-18.
[21] S. Smale, Morse theory and a non-linear generalization of the
Dirichlet problem, Ann. of Math. (2) 80 (1964), 382-396.

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Reference V[22] S. Smale, An inﬁnite dimensional version of Sard’s theorem, Amer. J.
Math. 87 (1965), 861-866.
[23] Y. Taketomi, On a Riemannian submanifold whose slice
representation has no nonzero ﬁxed points, Hiroshima Math. J. 48(2018), no. 1, 1-20.
[24] C.-L. Terng, Proper Fredholm submanifolds of Hilbert space. J.
Differential Geom. 29 (1989), no. 1, 9-47.
[25] C.-L. Terng, Polar actions on Hilbert space. J. Geom. Anal. 5 (1995),
no. 1, 129-150.
[26] C.-L. Terng, G. Thorbergsson, Submanifold geometry in symmetricspaces. J. Differential Geom. 42 (1995), no. 3, 665-718.

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101

Lagrangian submanifolds in complex projective
space and quaternionic Kähler geometry
Makoto Kimura
Abstract. We discuss a relationship of certain Lagrangian submanifolds incomplex projective space and submanifolds in complex 2-plane Grassmannian.

1. Let CPn be an n-dimensional complex projective space with Fubini-Study metricof constant holomorphic sectional curvature 4. For a Lagrangian submanifold M n in
CPn and a unit normal vector field N on M , we find the condition such that for r ∈ Rwith sufficiently small |r| > 0, 1-parameter family of ’parallel’ submanifolds
Mr := {expp (rNp )| p ∈ M }are Lagrangian submanifolds in CPn , by computing the differential of the normal exponential map (cf. [1]).
2. For a Lagrangian submanifold M n in CPn and a unit normal vector field N on
M , we define a ’Gauss map’ γ to complex 2-plane Grassmannian G2 (Cn+1 ) (cf. [2],[3], [4]). If M and N admits parallel Lagrangian submanifolds Mr as 1., and moreover
if JN is an eigenvector of the shape operator AN and the eigenvalue is constant, thenthe rank of γ is n − 1 and γ(M ) is a quarter dimensional totally real submanifold withrespect to both complex and quaternionic Kähler structure of G2 (Cn+1 ).
3. Conversely, let Σn−1 be a quarter dimensional submanifold in G2 (Cn+1 ) whichis totally real with respect to both complex and quaternionic Kähler structure ofG2 (Cn+1 ). Then we have locally a horizontal lift of Σ in the twistor space of G2 (Cn+1 ),and we can construct ruled (i.e., foliated by geodesics in CPn ) Lagrangian immersion ϕfrom a circle bundle M n over Σ to CPn , provided ϕ is regular, and M admits parallelfamily of Lagrangian submanifolds Mr in CPn .
This research was supported by JSPS KAKENHI Grant Number JP20K03575.
References[1] T. E. Cecil and P. J. Ryan, Focal sets and real hypersurfaces in complex projective space, Trans.
Amer. Math. Soc. 269 (1982), no. 2, 481–499.
[2] J. T. Cho and M. Kimura, Hopf hypersurfaces in complex hyperbolic space and submanifolds inindefinite complex 2-plane Grassmannian I, Topology Appl., 196 (2015), part B, 594–607.
[3] M. Kimura, Hopf hypersurfaces in complex projective space and half-dimensional totally complex
submanifolds in complex 2-plane Grassmannian I, Diff. Geom. Appl. 35, suppl., 266–273, II,
ibid. 54, part A, 44–52,[4] B. Palmer, Hamiltonian minimality and Hamiltonian stability of Gauss maps, Diff. Geom. Appl.
7 (1997), no. 1, 51–58.
(M. Kimura) Department of Mathematics, Faculty of Science, Ibaraki University,
Mito, Ibaraki 310-8512, Japan
E-mail address: makoto.kimura.geometry@vc.ibaraki.ac.jp

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Lagrangian submanifolds in complex
projective space and quaternionicKähler geometry.
Makoto Kimura(Ibaraki University)
March 5, 2021

Submanifolds of Symmetric Spaces and Their Timeevolutions

Makoto Kimura(Ibaraki University)

Lagrangian submanifolds

Contents’Parallel family’ of Lagrangian submanifolds inCPn ,Makoto Kimura(Ibaraki University)

Lagrangian submanifolds

Submanifolds of Symmetric Spaces and Their Time Evolutions

Contents’Parallel family’ of Lagrangian submanifolds in
CPn ,’Gauss map’ from Lagrangian submanifolds in
CPn to complex 2-plane Grassmannian
G2 (Cn+1 ),Makoto Kimura(Ibaraki University)

Lagrangian submanifolds

Contents’Parallel family’ of Lagrangian submanifolds in
CPn ,’Gauss map’ from Lagrangian submanifolds in
CPn to complex 2-plane Grassmannian
G2 (Cn+1 ),’Totally real’ submanifolds in G2 (Cn+1 ),Makoto Kimura(Ibaraki University)

Lagrangian submanifolds

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Contents’Parallel family’ of Lagrangian submanifolds in
CPn ,’Gauss map’ from Lagrangian submanifolds in
CPn to complex 2-plane Grassmannian
G2 (Cn+1 ),’Totally real’ submanifolds in G2 (Cn+1 ),Examples.

Makoto Kimura(Ibaraki University)

Lagrangian submanifolds

Parallel hypersurfaces.
Parallel hypersurfaces..
f : Riemann manifold,Mf,
M : An oriented hypersurface of Mf,
Nx : Unit normal vector ﬁeld M in M
ϕr (M ) := {expx (rNx )| x ∈ M } (0 < r)is called a parallel hypersurface of M , providedfϕ. r (M ) is a smooth hypersurface of M .
Makoto Kimura(Ibaraki University)

Lagrangian submanifolds

Submanifolds of Symmetric Spaces and Their Time Evolutions

Examples of parallel hypersurfaces in R3:
Circular cylinders

Makoto Kimura(Ibaraki University)

Lagrangian submanifolds

Examples of parallel hypersurfaces in R3:
tori of revolution

Makoto Kimura(Ibaraki University)

Lagrangian submanifolds

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Isoparametric hypersurfaces in space forms.
Isoparametric hypersurfaces in space forms..
If M0n : is an isoparametric hypersurface (i.e.,
f n+1 (c) ofprincipal curvatures are constant) in M
constant sectional curvature c, then its parallelf n+1 (c) is also
hypersurfaces Mrn in M
isoparametric.
.

Makoto Kimura(Ibaraki University)

Lagrangian submanifolds

Hopf hypersurfaces in Kähler manifold.
Structure vector of real hypersurface..
M 2n−1 : Oriented real hypersurface in Kählerfn,
manifold Mf,
N : unit normal vector ﬁeld of M in M
.ξ := −JN : structure vector ﬁeld of M .
.
Hopf hypersurface..
fn,
M 2n−1 : real hypersurface in Kähler manifold M
f
M : Hopf hypersurface in M.⇔ Aξ = µξ (µ: Hopf principal curvature) .
f.
A: shape operator of M in M
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Hopf hypersurfaces in complex space forms.
Hopf principal curvature..
f=M
f n (c) with constantµ is constant when M.holomorphic sectional curvature c ̸= 0.
.
Hopf hypersurface..
f n (c) (c ̸= 0)
M 2n−1 is a Hopf hypersurface in M⇒
f n (c) are also
Parallel hypersurfaces Mr in M
Hopf,
Each integral curve of ξ is a geodesic on M
f 1 (c) ⊂ M
f n (c).
and a ’circle’ in M.
Makoto Kimura(Ibaraki University)

Lagrangian submanifolds

Parallel family of Lagrangian submanifolds.
Parallel family of Lagrangian submanifolds in CPn..
In this talk, we treat with 1-parameter family of’parallel’ Lagrangian submanifolds Mrn in CPn with
Fubini-Study metric of constant holomorphic
sectional
curvature 4.
.

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Problem.
.Problem.
Let M0 be a Lagrangian submanifold in CPn and
let N be a unit normal vector ﬁeld on M . Forr ∈ R with suﬃcient small |r|, when each of theparallel family
Mr := {expp (rNp ) ∈ CPn | p ∈ M0 }

(1)

n
is
. a Lagrangian submanifold in CP ?

Makoto Kimura(Ibaraki University)

Lagrangian submanifolds

Answer.
.Answer.
For r ∈ R (with small |r|), each n-dimensional
submanifold Mr in CPn given by (1) is Lagrangian
if and only if the following 4 equations are satisﬁed:.. ∇JN (JN ) = 0, i.e., J N is a geodesic vector
1ﬁeld on M0 , where J and ∇ denote thecomplex structure and the induced Levi-Civita
connection on M0 , respectively,.. ⟨AN (JN ), ∇X (J N )⟩ = 0, where X, A
2and ⟨ , ⟩ denote a tangent vector ﬁeld
orthogonal to J N , the shape operator and theinduced metric of M0 , respectively,.

.
.

Makoto Kimura(Ibaraki University)

Lagrangian submanifolds

Submanifolds of Symmetric Spaces and Their Time Evolutions

Answer.
.Answer.
.. ⟨∇X (JN ), Y ⟩ = ⟨∇Y (JN ), X⟩, where X
3
and Y are tangent vector ﬁelds on M0 which
are orthogonal to J N , i.e., {J N }⊥ is an
integrable distribution on M0 ,.. ⟨∇X (JN ), AN Y ⟩ = ⟨∇Y (J N ), AN X⟩,
4
where X and Y are tangent vector ﬁelds on
M0 which are orthogonal to J N .
.

.
.

Makoto Kimura(Ibaraki University)

Lagrangian submanifolds

Integral curve of JN.
Integral curve of JN..
Furthermore we consider the case:.. JN is an eigenvector of the shape operator
5
AN , i.e., AN (J N ) = µJ N and theeigenvalue µ is constant on M0 .
Then each integral curve of JN in M0 is a circle in
a complex projective line CP1 in CPn . Moreover foreach r, with respect to a unit normal vector ﬁeld
(Nr )exp(rNp ) := (d exp)rNp (Np ) on Mr , each
integral curve of JNr in Mr is either a circle or ageodesic(ruled Lagrangian submanifold) in CPn .
.

.

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Complex Stiefel manifold.
Euclidean inner product on Cn+1..
For z, w ∈ Cn+1 , we deﬁne R-valued Euclidean∗.inner product by ⟨z, w⟩ = Re(z w).
.
Complex Stiefel manifold..
V2 (Cn+1 ) = {(u1 , u2 ) ∈ Cn+1 × Cn+1 |∥u1 ∥ = ∥u2 ∥ = 1, ⟨u1 , u2 ⟩ = ⟨u1 , iu2 ⟩ = 0}:Complex Stiefel manifold,n+1
M) = U (n + 1)/U (n − 1).
. V := V2 (CMakoto Kimura(Ibaraki University)

Lagrangian submanifolds

Complex 2-plane Grassmann manifold.
Complex 2-plane Grassmann manifold..
Complex 2-plane Grassmann manifold G2 (Cn+1 ) isrealized as a quotient space of an action of U (2) on
V2 (Cn+1 ) as U (2) ∋ g, (u1 , u2 ) 7→ (u1 , u2 )g.
n+1) = U (n + 1)/U (n − 1) × U (2).
.G2 (C

Makoto Kimura(Ibaraki University)

Lagrangian submanifolds

Submanifolds of Symmetric Spaces and Their Time Evolutions

Circle actions on V2(Cn+1).
Circle actions on complex Stiefel manifold..
We consider circle actions on MV :S 1 = R/2πZ ∋ θ, (u1 , u2 ) 7→ (eiθ u1 , eiθ u2 ),()
cos t − sin t
SO(2) ∋,
sin t cost()
cos t − sin t.
(u1 , u2 ) 7→ (u1 , u2 )
sin
t
cos
t
.

Makoto Kimura(Ibaraki University)

Lagrangian submanifolds

Sequence of projections.
Sequence of projections..
πSO(2)
πS 1
πZ
MV −
−
→ MS −−−→ MZ −→
MG ,
2
S
.
MG4n−4 : G2 (Cn+1 ), Complex 2-plane
Grassmannian,
MZ4n−2 : twistor space of MG , complex contact
manifold,
MS4n−1 : S 1 -bundle over MZ (3-Sasakian).
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Tangent spaces.
Tangent space of MV..
T(u1 ,u2 ) MV = ({u1 , u2 }⊥ × {u1 , u2 }⊥ )⊕R(iu1 , iu2 ) ⊕ R(−u2 , u1 )⊕R(−iu1 , iu2 ) ⊕ R(iu2 , iu1 ),.
.
Tangent space of MS..

.

T[u1 ,u2 ] MS ∼= ({u1 , u2 }⊥ × {u1 , u2 }⊥ )⊕R(−u2 , u1 )⊕R(−iu1 , iu2 ) ⊕ R(iu2 , iu1 ),Makoto Kimura(Ibaraki University)

Lagrangian submanifolds

Tangent spaces.
Tangent space of MZ..
T[u1 ,u2 ] MZ ∼= ({u1 , u2 }⊥ × {u1 , u2 }⊥ )⊕R(−iu1 , iu2 ) ⊕ R(iu2 , iu1 ),.
.
Tangent space of MG..
.

T[u1 ,u2 ] MG ∼= ({u1 , u2 }⊥ × {u1 , u2 }⊥ ).

Makoto Kimura(Ibaraki University)

Lagrangian submanifolds

Submanifolds of Symmetric Spaces and Their Time Evolutions

Q.K. structure of MG.
structure of MG.Q.K.
.
Then for (u1 , u2 ) ∈ V2 (Cn+1 ), a basis I1 , I2 , I3
of quaternionic Kähler structure of MG at [u1 , u2 ]is give as follows:.

(x1 , x2 ) ∈ {u1 , u2 }⊥ × {u1 , u2 }⊥ ,I1 : (x1 , x2 ) 7→ (−x2 , x1 ),I2 : (x1 , x2 ) 7→ (−ix1 , ix2 ),I3 : (x1 , x2 ) 7→ (ix2 , ix1 ).

Complex structure is J : (x1 , x2 ) 7→ (ix1 , ix2 ).
Makoto Kimura(Ibaraki University)

Lagrangian submanifolds’Gauss map’ to G2(Cn+1).
’Gauss map’ to G2 (Cn+1 )..
Let M0n be a Lagrangian submanifold in CPn and
let N be a unit normal vector ﬁeld on M0 . Then
we have a ’Gauss map’ γN from M0 to thecomplex 2-plane Grassmannian G2 (Cn+1 ), whereγN (p) is the complex 2-plane spanned by the
position
vector p and Np in Cn+1 for p ∈ M n .
.

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Theorem 1.
1.Theorem.
Let M n be a Lagrangian submanifold in CPn and
let N be a unit normal vector ﬁeld on M . Supposethat for each r ∈ R with suﬃciently small |r|, eachof the parallel family Mr of M0 is a Lagrangiansubmanifold in CPn and J N is an eigenvector of
the shape operator AN of M0 with constanteigenvalueµ.
.

Makoto Kimura(Ibaraki University)

Lagrangian submanifolds

Theorem 1.
1.Theorem.
Then.. The image γN (M0 ) is a quarter-dimensional
1
totally real submanifold with respect to bothcomplex structure and Quaternionic Kähler
structure of G2 (Cn+1 ), and.. Each ﬁber of γN is an integral curve of JN
2
on M0 .
.. Hence M0 is a total space of circle bundle over3
γN (M0 ) with projection γ.
.

.
.
.

Makoto Kimura(Ibaraki University)

Lagrangian submanifolds

Submanifolds of Symmetric Spaces and Their Time Evolutions’Gauss map’ of real hypersurfaces.
’Gauss map’ of real hypersurfaces..
Similar result was obtained for real hypersurfaces, inparticular Hopf hypersurfaces in CPn andhalf-dimensional totally complex submanufolds in
n+1
G), [K, 2014].
. 2 (C

Makoto Kimura(Ibaraki University)

Lagrangian submanifolds

Gauss map of hypersurface in sphere.
Gauss map of hypersurface in Sn+1..
x : M n → Sn+1 : an immersion,
N : M n → Rn+2 : unit normal vector ﬁeld,
e 2 (Rn+2 ) ∼γ : Mn → G= Qn : Gauss map, deﬁned.by γ(p) = x(p) ∧ N (p).
Here M n is an oriented hypersurface in Sn+1 ,e 2 (Rn+2 ): oriented real 2-plane Grassmannian,G
Qn : complex hyperquadric in CPn+1 .
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Gauss map of hypersurface in sphere.
Gauss map of hypersurface in Sn+1..
x : M n → Sn+1 : an immersion,
N : M n → Rn+2 : unit normal vector ﬁeld,
e 2 (Rn+2 ) ∼γ : Mn → G= Qn : Gauss map, deﬁned.by γ(p) = x(p) ∧ N (p).
Here M n is an oriented hypersurface in Sn+1 ,e 2 (Rn+2 ): oriented real 2-plane Grassmannian,G
Qn : complex hyperquadric in CPn+1 .
Makoto Kimura(Ibaraki University)

Lagrangian submanifolds

Gauss map of hypersurface in sphere.
Theorem (Palmer, 1997)..
M n ; oriented hypersurface in Sn+1⇒γ(M ): Lagrangian submanifold in Qn , andM n is either isoparametric or austere in Sn+1⇒γ(M): Minimal Lagrangian in Qn .
.
For t ∈ R, Mt := {cos tx(p) + sin tN (p)}:parallel hypersurface of M , the Gauss image is notchanged:γ(Mt ) = γ(M ).
Makoto Kimura(Ibaraki University)

Lagrangian submanifolds

Submanifolds of Symmetric Spaces and Their Time Evolutions

Proposition 2.
Proposition 2..
Let φ : Σn−1 → MG be a totally real immersion
concerning both complex structure and quaternionice the universal covering ofKähler structure of MG , Σ
e → Σ the covering projection. ThenΣ, and πΣ : Σ
e → MZe:Σ
there exists a horizontal immersion φ
e = φ ◦ πΣ (φ:e horizontal lift ofsuch that πZ ◦ φφ).
.

Makoto Kimura(Ibaraki University)

Lagrangian submanifolds

Converse construction.
construction.Converse.
Let ψ : Σn−1 → MZ be a horizontal immersion
such that πZ ◦ ψ : Σ → MG is a totally real
immersion concerning both complex structure andquaternionic Kähler structure of MG .
Concerning a circle bundle MS → MZ , letπψ : ψ ∗ MS → Σ be the pullback bundle over Σ
for
. ψ.

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Converse construction.
construction.Converse.
Then we have the following commutative diagram:ψ ∗ MS −−→ MS
η



π
πψ y
y Z
.

Σn−1 −−→ MZ .
ψ

Makoto Kimura(Ibaraki University)

Lagrangian submanifolds

Converse construction.
construction.Converse.
For a projection pr : MS → CPn ,[u1 , u2 ] 7→ [u1 ], We deﬁne a map
Φ0 := pr ◦η : ψ ∗ MS → CPn . Then the image ofeach ﬁber π −1 (p) for p ∈ Σ is a geodesic cp in
CPn and on the open subset O ⊂ ψ ∗ MS of
regular points for Φ0 , Φ0 (O) is a ruled Lagrangiansubmanifold
in CPn .
.

Makoto Kimura(Ibaraki University)

Lagrangian submanifolds

Submanifolds of Symmetric Spaces and Their Time Evolutions

Converse construction.
construction.Converse.
For a ruled Lagrangian submanifold M0 = Φ0 (O),along geodesics normal to M0 with initial vector
J ċp , each of parallel family of n-dimensional
submanifold Mr in CPn is also Lagrangian for
suﬃciently
small |r|.
.

Makoto Kimura(Ibaraki University)

Lagrangian submanifolds

Special case.
Special case..
Complex quadric Qn−1 in CPn which is identiﬁed
e 2 (Rn+1 )with real oriented 2-plane Grassmannian G
is half-dimensional totally geodesic, totally complex(concerning Q.K. structure) and totally real(concerning complex structure) submanifold in
n+1G).
. 2 (C

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Gauss image of hypersurfaces in S n.
Special case..
Hence Gauss image of oriented hypersurface Σn−1
in S n is considered as a totally real submanifold
concerning both complex structure and Q.K.
structure in G2 (Cn+1 ), so it satisﬁes conditions toconstruct parallel family of Lagrangian submanifoldsnM. r in CP .

Makoto Kimura(Ibaraki University)

Lagrangian submanifolds

Examples.
Examples..
Let Σn−1 be an oriented hypersurface, p a position
vector of Σ in S n ⊂ Rn+1 and Np a unit normalof Σn−1 in S n at p ∈ Σ.
For r ∈ R with suﬃciently small |r|, we deﬁne amap, Φr : M n = S 1 × Σn−1 → CPn ,Φr (t, p) := π((cos r cos t − i sin r sin t)p+(− sin r cos t + i cos r sin t)Np ),where
π : S 2n+1 → CPn is the Hopf ﬁbration.
.
Makoto Kimura(Ibaraki University)

Lagrangian submanifolds

Submanifolds of Symmetric Spaces and Their Time Evolutions

Examples.
Examples..
Then {Φr (M n )} gives a parallel family of
Lagrangian submanifold in CPn , provided Φr is an
immersion. Here t is a parameter of integral curve(a circle lies in CP1 ) of JN .
In particular when r = 0, Φ0 (M ) is a ruled
Lagrangian submanifold in CPn , provided the shapen−1
in S n is non-degenerate.
.operator of Σ

Makoto Kimura(Ibaraki University)

Lagrangian submanifolds

References.
.References.
.. D. V. Alekseevsky and S. Marchiafava, A
1
twistor construction of Kähler submanifolds ofa quaternionic Kähler manifold, Ann. Mat.
Pura Appl. 184 (2005), no. 1, 53–74.
.. J. T. Cho and M. Kimura, Hopf hypersurfaces
2
in complex hyperbolic space and submanifolds
in indefinite complex 2-plane Grassmannian I,
Topology Appl., 196 (2015), part B, 594–607.
.

.
.

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References.
.References.
.. M. Kimura, Hopf hypersurfaces in complex
3
projective space and half-dimensional totallycomplex submanifolds in complex 2-plane
Grassmannian I, Diﬀ. Geom. Appl. 35, suppl.,
266–273, II, ibid. 54, part A, 44–52,.. B. Palmer, Hamiltonian minimality and4
Hamiltonian stability of Gauss maps, Diﬀ.
Geom. Appl. 7 (1997), no. 1, 51–58.
.

.

.

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An introduction to the deformed HermitianYang-Mills (dHYM) connections
Hikaru Yamamoto
Abstract. In this talk, I gave an introduction to the deformed Hermitian Yang–
Mills (dHYM) connections. The talk was started with the introduction of somebasic notions of special Lagrangian submanifolds. After that, the easy version of
the real Fourier–Mukai transform was explained. Finally, I gave a list of recent
results obtained in a joint works with K. Kawai.

1Special Lagrangian submanifold

Roughly speaking, a deformed Hermitian Yang–Mills connection is a mirror object ofa special Lagrangian submanifold in the sense of mirror symmetry. So, it’s better tostart with the introduction of special Lagrangian submanifolds.
iy. Denote
Let Cn ∼= R2n be the standard complex plane with coordinates x = x +Pnthe standard Kähler form and the holomorphic volume form on C by ω = i dxi ∧ dy iand Ω := dz 1 ∧ · · · ∧ dz n , respectively.
Definition 1. An n-dimensional real submanifold L in Cn is called a special Lagrangiansubmanifold with phase eiθ if it satisfies ω|L = 0 and Im(e−iθ Ω)|L = 0.
It is well-known that a special Lagrangian submanifold is volume minimizing in its
homology class. One can easily see that if a Lagrangian submanifold L ∈ Cn is writtenby the graph of the gradient of some function f on Rn the special Lagrangian conditionis equivalent toarg det (I + i Hess f ) = θ.
(1)
It is also known (as a result of McLean) that the moduli space of special Lagrangiansubmanifolds (around L) is a smooth b1 (L)-dimensional manifold.

2

The real Fourier–Mukai transform

Historically, the definition of deformed Hermitian Yang–Mills connections is introducedby the real Fourier–Mukai transform in the paper in the paper of Leung, Yau andZaslow. To explain that, let f : Rn → R be a smooth function. Then, as explained
above, we obtain the graph of Y := ∇f denoted by SY . On the other hand, we also
get a connection bynXY∇ := d +Y i dy i .
i=1This is an Hermitian connection of the trivial C-bundle over Cn . Then, by Leung, Yau
and Zaslow, it was found that f satisfies (1) if and only if the curvature 2-form F∇YsatisfiesIm e−iθ (ω + F∇Y )n = 0.

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We remark that the condition that Y is written as ∇f is equivalent to F∇0,2Y = 0. Thus,
we have reached the definition of deformed Hermitian Yang–Mills connections.
Definition 2. Let (X, ω) be a Kähler manifold with dimC X = n and L → X be asmooth C-bundle with an Hermitian metric h. Then, an Hermitian connection ∇ of(L, h) is called a deformed Hermitian Yang–Mills connection with phase eiθ if it satisfies
F∇0,2 = 0 andIm e−iθ (ω + F∇Y )n = 0,
where F∇ is the curvature 2-form of ∇.
The correspondence between SY and ∇Y is (roughly) called the real Fourier–Mukai
transform.

3

Some results

We are wondering whether some properties of special Lagrangian submanifolds also
hold for deformed Hermitian Yang–Mills connections or not. In other words, we are
wondering whether deformed Hermitian Yang–Mills connections are similar to special
Lagrangian submanifolds or not. This is a motivation of a joint work with K. Kawai.
We answered the following questions affirmatively.
• Is a deformed Hermitian Yang–Mills connection a minimizer of some functional?
• Is there a flow similar to mean curvature flows?
• Is the moduli space of deformed Hermitian Yang–Mills connections smooth andfinite dimensional?
We give some more detail on the first and second question. First, the functional V
on the set of all Hermitian connections of (L, h) is given by
Z r

 ωn
det idT X − iF∇].
V (∇) :=n!
X
We call V the volume functional for Hermitian connections. As the case of submanifolds, we can define a 1-form depends on ∇, denoted by H(∇), which satisfies the firstvariation formula:Zδ∇ V = − h · , H(∇)i.
X

Then, the first main theorem of a joint work with Kawai is as follows .
Theorem 1. Assume that (X, ω) is a Kähler manifold with dimC X = 3 or 4. Then,the functional V has a topologically determined lower bound and it is attained at ∇ if
and only if it is a deformed Hermitian Yang–Mills connection.

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As the case of mean curvature flows, we can consider the negative gradient flow of
V and it written as∂∇ = H(∇).
(2)∂t
Of course, the volume functional V is decreasing along this flow. So, we can expectthat the flow converges a deformed Hermitian Yang–Mills connection if the flow has
a long time solution. As the first step to realize this expectation, we have proved thefollowing, the short time existence and uniqueness.
Theorem 2. Assume that X is compact. Then, the flow (2) satisfies the short time
existence and uniqueness for any initial connection ∇.
(H. Yamamoto) Department of Mathematics, Faculty of Pure and Applied Science,
University of Tsukuba, 1-1-1 Tennodai, Tsukuba, Ibaraki 305-8577, Japan
E-mail address: hyamamoto@math.tsukuba.ac.jp

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Line bundle mean curvature flows and the moduli
space of dHYM connections
Hikaru Yamamoto
Abstract. In this talk, I introduced two recent results. One is given in a joint
work with X. Han. It is about an ε-regularity theorem for line bundle mean
curvature flow. The other is given in a joint work with K. Kawai. It proves thatthe moduli space of deformed Hermitian Yang–Mills connections is a smooth finite
dimensional manifold.

1

Line bundle mean curvature flow

In the former talk, I introduced a flow for Hermitian connections. There is a metricversion of the flow, the so-called line bundle mean curvature flow. It was introduced
by Jacob and Yau.
Definition 1. Let (X, ω) be a Kähler manifold with dimC = n and L → X be a holomorphic line bundle. Then, a one-parameter family of Hermitian metrics { ht }t∈[0,T )of L is called a line bundle mean curvature flow if it satisfies∂(− log ht ) = Θ(ht ) − θ,∂for some fixed constant θ ∈ R.
In the above definition, the angle function Θ(h) is defined to satisfy(ω − ∂ ∂¯ log h)n = R(h)eiΘ(h) ω n .
The function R(h) is called the radius function.
The line bundle mean curvature flow is considered as a mirror object of a mean
curvature flow. So, it could be singular in finite time as the mean curvature flow case.
Thus, I considered that establishing an ε-regularity theorem for line bundle mean
curvature flows will be useful.
To introduce our main result, we should put some assumptions. Assume that X
is diffeomorphic to the product of balls B(r1 ) and B(r2 ) in Rn and L is the trivial
C-bundle. Also assume that the Kähler form and ht are y-invariant. Then, we definea kind of “the Gaussian density” for Hermitian connections byZ|x|2 + |∂ log ht |2
ωn
1
−1
exp
−
R(h
)
.
D(ht ) := (Vol(B(r2 )))t(4π(T − t))n/2 X
4(T − t)n!
Then, the ε-regularity theorem given in a joint work with Han is expressed asfollows.
Theorem 1. There exist C > 0 and ε > 0 satisfying the following condition. If a line
bundle mean curvature flow { ht }t∈[0,T ) satisfies|F (ht )| ≤ Cthen ht can be extended beyond T .

andlim D(ht ) < 1 + ,

t→T

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The moduli space of dHYM connections

Next, we switch the subject to connections from metrics. So, we consider deformed
Hermitian Yang–Mills connections, not metrics. This part is based on a joint work with
K. Kawai. We studied the moduli space of deformed Hermitian Yang–Mills connections.
The setting is as follows. Let (X, ω) be a compact Kähler manifold and let L → X
be a smooth C-bundle with an Hermitian metric h. Fix θ ∈ R. Put
M := { dHYM connections of (L, h) }/U (1)-gauge
Then, we proved the following.
Theorem 2. If M 6= ∅, then M is a smooth b1 (X)-dimensional manifold. Moreover,it has an affine structure and it is orientable.
(H. Yamamoto) Department of Mathematics, Faculty of Pure and Applied Science,
University of Tsukuba, 1-1-1 Tennodai, Tsukuba, Ibaraki 305-8577, Japan
E-mail address: hyamamoto@math.tsukuba.ac.jp

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Lagrangian mean curvature flows with generalized
perpendicular symmetries
Akifumi Ochiai
Abstract. We show a method of constructing an invariant Lagrangian mean
curvature flow in a Calabi-Yau manifold with the use of generalized perpendicularsymmetries. We use moment maps of the action of Lie groups, which are notnecessarily abelian. We also show a general way to construct an invariant mean
curvature flow in a Riemannian manifold.

1

Preliminaries

Definition 1. Let φ : Σ → M be an immersion from a manifold Σ to a manifold M .
For a smooth map(
F : Σ × [0, T ) → M ; (p, t) 7→ Ft (p),F0 = φif Ft (·) : Σ → M is an immersion for any t ∈ [0, T ), then we call F a deformation of φ.
Definition 2. Let φ : Σ → (M, g) be an immersion from a manifold Σ to a Riemannian
manifold (M, g). The mean curvature flow F = (Ft )t∈[0,T ) of φ is the deformation of φsuch that it is a smooth solution of the following partial differential equation for the∂mean curvature vector field Ht of Ft : ∂t
F (p, t) = Ht (p).

2Constructions of invariant mean curvature flows

Let (M, g) be a Riemannian manifold, H a Lie group which acts on M , K a closed
subgroup of H, Z(h∗ ) the center of the Lie coalgebra h∗ , Lh : M → M the translation
by an element h ∈ H, ξ # a fundamental vector field generated by ξ ∈ h =: Lie (H),
LK a subset in L defined by LK = {p ∈ L | Hp = K} for any submanifold L of M ,where Hp is the isotropy subgroup of H at p ∈ L, V a submanifold of M such thatV ⊂ M K , φV : (H/K) × V → M a map defined by (hK, p) 7→ hp. We also use thesenotations in the following section.
Definition 3. If the map φV is an immersion with the mean curvature vector field H
and it holds that
H(hK, p) = (Lh )∗p H(K, p) ((hK, p) ∈ (H/K) × V ),then we say that V has the property (∗). Moreover, if there exists a deformationf : V × [0, T ) → M K of V such that the immersed submanifold Vt := ft (V ) also has
the property (∗), we say that f preserves the property (∗) of V.
Definition 4. Suppose there exists a deformation f : V × [0, T ) → M K . If φVt is an
immersion for any t ∈ [0, T ), the following map F defines a deformation of φV0 and wesay F the expansion of f :F : (H/K) × V × [0, T ) → M ;(hK, p, t) 7→ hft (p).

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Theorem 1. Suppose a submanifold V ⊂ M K has the property (∗) and there exists a
deformation f : V × [0, T ) → M K of V with the expansion F satisfying the followings.
∂
Ft (K, p) = Ht (K, p), and(i) For any t ∈ [0, T ) and any p ∈ V , it holds that ∂t(ii) the deformation f preserves the property (∗),
where, Ht is the mean curvature vector field of the immersion Ft . Then, the family ofmaps (Ft )t∈[0,T ) is the mean curvature flow of the map φV .
Corollary 1. Suppose that a submanifold V ⊂ M K has the property (∗) and there
exists a vector field A satisfying the followings.
(i.a) The vector field A generates a deformation f : V × [0, T ) → M K of V with the
expansion F ,(i.b) for any t ∈ [0, T ) and any p ∈ V , it holds that Ht (K, p) = Aft (p) , and(ii) the deformation f preserves the property (∗).
Then, the family of maps (Ft )t∈[0,T ) is the mean curvature flow of the map φV .
We note that for a given submanifold V , if we find a vector field A satisfying thecondition (i.b), then the condition (i.a) is an ordinary differential equation.

3

Constructions of invariant Lagrangian mean curvature flows

We show a method of constructing a Lagrangian mean curvature flow in a Calabi-Yau
manifold by the corollary above, using generalized perpendicular symmetries.
Theorem 2. Let (M, I, g, Ω) be a connected Calabi-Yau manifold with a complex structure I, Kähler metric g and a Calabi-Yau structure Ω, H a connected Lie group whichacts on M preserving I and g with a moment map µ : M → h∗ , K a closed subgroup of
H such that H/K is orientable and the K-action
 preserves Ω, L a special Lagrangiansubmanifold, c ∈ Z(h∗ ), Vc an n − dim(H/K) -dimensional submanifold of M suchthat Vc ⊂ µ−1 (c) ∩ LK and φVc is an immersion. Then, it holds that(1) the map φVc is a Lagrangian immersion, and(2) there exists a vector field AH along LK such that
Hc (hK, p) = (AH )hp + (Lh )∗p Ip (gradφ∗V g θc (K, ·))pc((hK, p) ∈ (H/K) × Vc )holds, where Hc is the mean curvature vector field of φVc and θc : (H/K) × Vc →R/2πZ is the Lagrangian angle of φVc .
Moreover, suppose that AH generates a deformation f : Vc × [0, T ) → LK with the
expansion F and for any t ∈ [0, T ) and Vt := ft (Vc ), the following condition holds.
#
ξp# ∈ T⊥
/ Tp Vc \{0}p L ⊕ Tp Vc and ξp ∈(“generalized perpendicular condition00 ).

Then, the family of maps (Ft )t∈[0,T ) is the Lagrangian mean curvature flow of φVc .

Submanifolds of Symmetric Spaces and Their Time Evolutions

137(A. Ochiai) Department of Mathematical Sciences, Tokyo Metropolitan University, 1-1 Minami-osawa, Hachioji-shi, Tokyo, 192-0397 Japan.
E-mail address: a-ochiai@tmu.ac.jp, akfm.oc@gmail.com

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Lagrangian mean curvature flows
with generalized perpendicular symmetries
Tokyo Metropolitan University, Akifumi Ochiai
March 6, 2021
Submanifolds of Symmetric Spaces and Their Time Evolutions

1§1. Goals•Goal（general cases）：
To construct mean curvature flows by symmetries of Lie groups inRiemannian mfds.
•Goal（special cases）：
To construct Lagrangian mean curvature flows by generalized perpendicular symmetries of Lie groups in Calabi-Yau mfds.

2

Submanifolds of Symmetric Spaces and Their Time Evolutions

139§2. Previous Researches•Previous Researches:
Yamamoto(2016)
constructinusing
Konno(2018)
constructinusing

generalized Lag MCF
toric almost Calabi-Yau mfds
moment map & toric symm.
Lag MCFCalabi-Yau mfds
moment map & perp. symm. of abelian actions•Our Researches:Ours(general cases)
construct
in
using
Ours
(special cases)
construct
in
using

MCF
Riem. mfds
symm. of general actions

Lag MCFCalabi-Yau mfds
moment map & generalized perp. symm. of general actions

3

-

§3.

Overview

1が凹

How to construct MCF by symm. of Lie groups
M : Riem. mfd, H: Lie grp s.t. H ! M,
Σ: H-invariant submfd of M.
Step.1 Find a nice sumfd V0 ⊂ Σ s.t. H · V0 = Σ.

Step.2 Study how V0 is deformed by the MCF of Σ.

&𩻛Step.3 Have the MCF of Σ by Σt := H · Vt.

4

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mfd

mfd

Def. 1 Let φ : Σ → M be an immersion. For a smooth map!
F : Σ × [0, T) → M; (p, t) $→ Ft(p),F0 = φif Ft(·) : Σ → M is an immersion for ∀t ∈ [0, T), then we call F thedeformation of φ (or Σ).
mfd

Riem.mfd

Let φ : Σ → (M, g) be an immersion. The mean curvature flow
F = (Ft)t∈[0,T) of φ is the deformation of φ s.t. it is a smooth solutionof the following PDE:∂
F(p, t) = H t(p)∂tw/ H t : mcv of Ft.

Fact. 2 MCF preserves the “Lagrangeness” in Kähler-Einstein mfds.
5§5.

Constructions of MCFs

•Setting (∗1):
·
·
·
·

(h)

y Hlkp )(M, g): Riem. mfd,
H: Lie grp s.t. H ! M,
K: closed subgrp of H,
V: submfd of M s.t. V ⊂ MK .

Def. 3 Under (∗1),

H※がHlhkip )

、

p.

hp

h

T

Z(h∗) : the center of the Lie coalgebra h∗,
LK := {p ∈ L | Hp = K},

φV : (H/K) × V → M;

w/L : any submfd of M,(hK, p) $→ hp.

Def. 4 (property (∗)) Under (∗1), if φV is an immersion & its mean
curvature vectors are H-invariant, i.e., it holds that(∗)

H(hK, p) = (Lh)∗pH(K, p),then we say that V has the property (∗) wrt the H-actions.
6Submanifolds of Symmetric Spaces and Their Time Evolutions

141

Def. 5 (preserve the property (∗))
Let V0 is a submfd of M s.t.
K
V0 ⊂ M & has the property (∗). Under (∗1), if ∃ a deformationof V0 in MK & Vt := ft(V) also has the property (∗), we say that f
preserves the property (∗) of V0.

7

Under (∗1), suppose that ∃ a deformation f : V0 × [0, T) → MK .

Def. 6 (expansion of deformation) If φVt is an immersion for ∀t ∈[0, T), we can define a deformation F of φV0 by
F : (H/K) × V0 × [0, T) → M;

(hK, p, t) '→ h ft(p) =: Ft(hK, p).

We call F the expansion of f .
We denote the mean curvature vector of Ft by H t.

か

な塔

虛

䵷 :藦
彝桜
→

F

.

.

.

た

な

を

-_-

声
Im

。

8

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i://.ie?!:LM癭欝
i.it:0?!!I!:蕺
iii.鰆竈iiiii
・•Setting (∗2):
·
·
·
·

H"

(M, g): Riem. mfd,
H: Lie grp s.t. H ! M,
K: closed subgrp of H,
V0: submfd with (∗) of M ( s.t. V0 ⊂ MK ).

で来Thm. 7 Under (∗2), suppose that ∃ a deformation f of V0 with its
expansion F satisfying (i) & (ii):
(i)

(ii)

For ∀t ∈ [0, T), ∀p ∈ V0,∂
Ft(K, p) = H t(K, p) (“restricted MCF condition))),∂tf preserves the property (∗).

Then, (Ft)t∈[0,T) is the MCF of φV0 .

9e.g. 8 (circle, sphere)
φ : Sn → Rn+1, V0 := single point, H := SO(n + 1).

H-actionve.g. 9 (cylinder)
φ : Sm × Rn−m → Rn+1, V0 := Sm, H := Rn−m.
八H-actionv

10Submanifolds of Symmetric Spaces and Their Time Evolutions

143

e.g. 10 (generalized cylinder)
φ : Mm × Rn−m → Rn+1, V0 := M, H := Rn−m.
八H-actionv

11

Question: How to reduce the restricted MCF eq to an ODE ?
! Additional assumption:
The evolution of the restricted MCF forms a vector field of the mean
curvature vectors.
e.g. 11(1) The MCF of Sn forms a vector filed of their mcv.

V0

⇧

:

滅簸簿

(2) The MCF of Dumbbell-like surfaces do not.

V0

✖

○

○

○

○

⇧

獷恭
鰆
翦 た
来
.to
i.

12

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OCAMI Reports Vol. 2 (2021)

Cor. 12 Under (∗2), suppose that the restricted MCF of (V0, φV0 )forms
a vector filed A, i.e., ∃ a vector field A satisfying (i.a) & (i.b):(i.a) A generates a deformation f of V0 in MK with F, i.e.,dFt(K, p) = A ft(p) (∀p ∈ V0, ∀t ∈ [0, T)) ← ODE
dt(i.b) For ∀t ∈ [0, T) & p ∈ V0,
Moerover, suppose that(ii)

H t(K, p) = A

f preserves the property (∗).

Then, (Ft)t∈[0,T) is the MCF of φV0 .

Eft (p) .

M

ii! How to find V0 with A satisfying (i.b) for constructing Lag MCFs
in CY mfds ?
14§6.

Constructions of Lag MCFs

•Setting (∗3):
·
·
·
·
·

(M, ω): 2n-dimR symp. mfd,
H: Lie grp s.t. H ! (M, ω) with moment map µ : M → h∗,
K: closed subgrp of H,
Vc: submfd of M s.t. Vc ⊂ MK ,φVc : immersion.

Prop. 13 Under (∗3), suppose(i) Vc is isotropic,(ii) (“moment map condition”) Vc ⊂ µ−1(c) for c ∈ Z(h∗).
(iii) dimH/K + dimVc = n
Then φVc is Lagrangian. Conversely, if φVc is connected & Lagrangian, then (i), (ii) and (iii) hold.

15Submanifolds of Symmetric Spaces and Their Time Evolutions

145

Def. 14 (Lagrangian angle) (M, I, g, Ω): Calabi-Yau mfd, L: oriented
Lag submafd of M,√

θ : L → R/2πZ : Lagrangian angle :⇔ ι∗Ω = e −1θvolι∗ g

w/ ι : L → M: inclusion map.

L : special Lagrangian submfd :⇔ θ ≡ const.
Prop. 15 H(p): mean curvature vector of L at p ∈ L. Then,!
"
H(p) = Iι(p) ι∗p(gradι∗ gθ)p .

16•Setting (∗4):· (M, I, ω, Ω): connected Calabi-Yau mfd,
· H: connected Lie grp s.t. H ! (M, I, ω)with moment map µ : M → h∗,· K: closed subgrp of H s.t. H/K: orientable & K ! Ω,· Vc: orientable submfd of M s.t. Vc ⊂ µ−1(c) ∩ MK ,· φVc : Lag immersion.
Prop. 16 Under (∗4),(1) θc(hK, p) = ∃ θH (hK) + ∃ θVc (p), w/ θc: Lag angle of φVc ,↑ defined only by! (M, H, K)"c(2) H (hK, p) = ∃ (AH )hp + (Lh)∗pIp (gradφ∗ gθVc )p ,Vc

↑ defined only by (M, H, K)
w/ H c :MCV of φVc .

If θVc ≡ const. ! H c = AH holds and Vc accomodates to Cor.12
20

146

OCAMI Reports Vol. 2 (2021)•Setting (∗5):· (M, I, ω, Ω): connected Calabi-Yau mfd,
· H: connected Lie grp s.t. H ! (M, I, ω)with moment map µ : M → h∗,· K: closed subgrp of H s.t. H/K: orientable & K ! Ω,
· AH : vector field along MK as in Prop.16· L: special Lag submfd with Lag angle θ(p) ≡ θ,∗
· c ∈ Z(h# ),$· Vc: n − dim(H/K) -dim submfd of M s.t. Vc ⊂ µ−1(c) ∩ LK .
Prop. 17 Under (∗5), suppose#∀p ∈ Vc, ∀ξ ∈ h, ξ#p ∈ T⊥
p L⊕Tp Vc & ξp ! Tp Vc \{0}.
(“generalized perp. condition”)

i

Then,
(1)
(2)

ン、

generalized>」

strictly perpθVc (p) = θ − π2 dim(H/K),
H c(hK, p) = (AH )hp.

← const.

21

Thm. 18 Under (∗5), suppose that AH generates a deformation f :Vc × [0, T) → LK with its expansion F, and for ∀t ∈ [0, T) and Vt :=ft(Vc), the generalized perpendicular condition holds.
Then, AH and Vc satisfies the condition of Cor.12 and (Ft)t∈[0,T) is a
Lag MCF of φVc .

レ、 AH
^

、

p

-_-

-__
_

、

ただ

-_.

.

.

.

Im帳

22

Submanifolds of Symmetric Spaces and Their Time Evolutions§7.

Examples147

がe.g. 19

constructinusing

Lag self-similar solution
C4
strictly perp. symm. of U(1) × SO(3)e.g. 20

construct
inusing

Lag MCFC5
gen. perp. symm. of R × SO(2)e.g. 21

construct
inusing

Lag translating soliton
C5strictly perp. symm. of U(1) × SO(3)e.g. 22

construct
inusing

Lag translating soliton
C6gen. perp. symm. of R × SO(2)An

23

Thank you very much for your attention.

24

、

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Ricci flow, heat equation, Liouville type theorem
Keita Kunikawa
Abstract. In this talk, we will see a Liouville type theorem for heat equation
along ancient (super) Ricci flow using Perelman’s reduced distance.

1. Liouville theorems
Let (M n , g) be a complete Riemannian manifold with Ric ≥ −K (K ≥ 0) and BR (x0 )be a geodesic ball on M centered at x0 ∈ M with radius R > 0. In 1975, Cheng-Yau
established a gradient estimate for positive harmonic function u : BR (x0 ) → R;
|∇u|
≤ Cn
u



1 √
+ K
R


on B R (x0 ),2

where Cn is a constant depending only on n. When K = 0, Cheng-Yau’s gradientestimate implies the celebrated Yau’s Liouville theorem, that is, positive harmonic
function on a complete Riemannian manifold with Ric ≥ 0 must be constant.
A parabolic analogue of this theory is considered by Souplet-Zhang [2]. They obtained a space-only gradient estimate for a positive solution to the heat equation ona Riemannian manifold with Ric ≥ −K. More precisely, they showed that for apositive solution to ∂t u = ∆u, 0 < u ≤ A on a parabolic cylinder QR,T (x0 , t0 ) =BR (x0 ) × [t0 − T, t0 ],
|∇u|
≤ Cn
u



√
1
1
+√ + KRT
A
1 + loguon Q R , T (x0 , t0 ).
2

4

As a corollary, they showed Liouville type results for ancient solutions (i.e., solutions
defined for all the negative time t ∈ (−∞, 0]) to heat equation on a complete Riemannian manifold with Ric ≥ 0;(a1) positive ancient solution u with u(x, t) = exp [o(dg (x) +be constant,(b1) ancient solution u with u(x, t) = o(dg (x) +pp|t|)] near infinity must|t|) near infinity must be constant.

Without the growth conditions, there exist nontrivial ancient solutions. For (a1), we
know a positive ancient solution u(x, t) = ex+t on M = R which does not satisy thegrowth condition. Likewise, we have another example for (b1): u(x, t) = x. This is astatic ancient solution on M = R which does not satisfy the growth condition. In thissense, Souplet-Zhang’s growth conditions are essential and sharp in the space direction.

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149

2. Main results
In this talk, we will see that the analogous results to Souplet-Zhang hold under Ricci
flow background. Althohg it is natural to consider the time-dependent Riemannian
distance dg(t) (x), one finds that it does not work well. The problem is that the condition
Ric ≥ −K is not sufficient to estimate the time derivative of dg(t) (x). If we additionallyimpose Ric ≥ −K, then it is possible to estimate the time derivative of dg(t) (x) and
we can obtain a gradient estimate as above. However, in order to derive Liouville type
results, this situation can deal with only trivial situation, i.e., Ricci flat (K = 0).
To overcome this difficulty, we adobt Perelman’s reduced geometry. In reduced
geometry, it is natural to use reverse time parameter τ := −t. Instead of dg(t) (x), we
adopt the so-called reduced distance `(x, τ ). This makes it possible for us to avoid theestimate of time-derivative of dg(t) (x).
Here, we will consider more general situation than the Ricci flow. Let (M, g(τ )) bea time-dependent Riemannian manifold, τ ∈ [0, ∞). Set h := 12 ∂τ g and H := trg h. For(x, τ ) ∈ M × (0, ∞), let L(x, τ ) stand for the L-distance from a space-time base point(x0 , 0), i.e., the infimum of the so-called L-length over all curves γ : [0, τ ] → M with
γ(0) = x0 and γ(τ ) = x. We only consider the case that the infimum is achieved bya minimal L-geodesic. Then the reduced distance from (x0 , 0) and its squred root isdefined byp
1`(x, τ ) = √ L(x, τ ), d(x, τ ) = 4τ `(x, τ ).
2 τIn the static case of g(τ ) ≡ g, it holds that `(x, τ ) = dg (x)2 /4τ . Moreover, we introduce
the Müller quantity D(V ) and the trace Harnack quantity H(V ) for (time-dependent)
vector field V on M by
D(V ) := −∂τ H − ∆H − 2||h||2 + divh(V ) − 2g(∇H, V ) + 2Ric − 2h(V, V ),H
− 2g(∇H, V ) + 2h(V, V ).
H(V ) := −∂τ H −τ
Main Theorem. (K.-Sakurai [1]) Let (M, g(τ ))τ ∈[0,∞) be a complete ancient backward
super Ricci flow Ric ≥ h with D(V ) ≥ 0, H(V ) ≥ −H/τ and H ≥ 0 for any V . Then,(a2) positive ancient solution to ∂τ u = −∆g(τ ) u with u(x, τ ) = exp[o(d(x, τ ) +near infinity must be constant;(b2) ancient solution to ∂τ u = −∆g(τ ) u with u(x, τ ) = o(d(x, τ ) +must be constant.

√

√
τ )]

τ ) near infinity

Remark. For a static manifold with Ric ≥ 0, all the assumptions automatically holdand d(x, τ ) = dg (x). So, our theorem includes Souplet-Zhang’s Liouville type result.
As for the Ricci flow Ric = h with bounded nonnegative curvature operator Rm, again
all the assumptions are satisfied and we obtain a Liouville type result.

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References[1] K. Kunikawa and Y. Sakurai, Liouville theorem for heat equation along ancient super Ricci flow
via reduced geometry, preprint, arXiv:2005.04882.
[2] P. Souplet and Q.S. Zhang, Sharp gradient estimate and Yau’s Liouville theorem for the heat
equation on noncompact manifolds, Bull. London Math. Soc. 38 (2006), no. 6, 1045–1053.
(K. Kunikawa) Cooperative Fuculty of Education, Utsunomiya University, 350
Mine-machi, Utsunomiya, 321-8505, Japan
E-mail address: kunikawa@cc.utsunomiya-u.ac.jp

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Antipodal sets of compact symmetric spaces andpolars of compact Lie groups
Makiko Sumi Tanaka
Abstract. This presentation is based on the author’s collaboration with Hiroyuki Tasaki. In their former research, Tasaki and the author classified maximal
antipodal sets of some classical compact Riemannian symmetric spaces by using
their embeddings into connected compact Lie groups as polars. In order to continue the classification of maximal antipodal sets of other compact Riemanniansymmetric spaces, their realization as polars of disconnected compact Lie groups
is needed. In this presentation the author explain the recent research relating tothem.

Let M be a compact Riemannian symmetric space and sx denote the point symmetry at a point x in M . A subset A of M is called an antipodal set if sx (y) = y holds forany points x, y in A. The 2-number of M is the maximum of the cardinalities of antipodal sets. In the 1980’s Chen and Nagano introduced these notions and gave detailed
studies of the 2-numbers. In the past ten years there was progress on the research of
antipodal sets. Our interest has been shifted to maximal antipodal sets themselvesrather than their cardinalities. Tasaki and the author classified maximal antipodal
subgroups of some classical compact Lie groups G and gave their explicit descriptions(J. Lie Theory, 2017), and after that, they classified maximal antipodal sets of someclassical compact Riemannian symmetric spaces M (Differential Geom. Appl., 2020).
The basic principle is to make use of an embedding of M into G as a polar with respect
to the identity element and apply the classification of maximal antipodal subgroupsof G. In order to continue the classification of maximal antipodal sets for some otherclassical compact Riemannian symmetric spaces M , the realization of M as a polar of
a disconnected compact Lie group is needed. Chen-Nagano (Duke Math. J., 1978) andNagano (Tokyo J. Math., 1988) gave detailed studies of polars of connected compact
Riemannian symmetric spaces. Tasaki and the author studied polars of disconnectedcompact Lie groups in their submitting paper.
Let G be a compact Lie group and we denote by e the identity element of G. Thereexists a biinvariant Riemannian metric on G and G is a Riemannian symmetric spacewith respect to the metric. For any x ∈ G, the point symmetry at x, denoted bysx , is given by sx (y) = xy −1 x (y ∈ G). Let G be a connected compact Lie groupand σ be an involutive automorphism of G. We denote by hσi the subgroup of thegroup of automorphisms of G generated by σ. For the semidirect product G o hσi,
G o hσi = (G, e0 ) ∪ (G, σ) is the decomposition into a disjoint union of the connectedcomponents, where e0 = idG denotes the identity element of hσi. We denote by ê theidentity element of the semidirect product G o hσi. We define the action ρσ of G on
G by ρσ (g)(h) = ghσ(g)−1 (g, h ∈ G), which is called the twisted conjugate action by
σ. For an isometry f of G, we denote by F (f, G) the set of fixed points of f . The
following is the main theorem:

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Theorem (Tanaka-Tasaki, submitted). Let G be a connected compact Lie groupand σ be an involutive automorphism of G. Then we obtain
F (sê , G o hσi) = (F (se , G), e0 ) ∪ (F (se ◦ σ, G), σ).
In particular, each connected component of (F (se ◦ σ, G), σ) is a polar of G o hσi.
Moreover, the connected component of (F (se ◦ σ, G), σ) containing (e, σ) coincides
with (ρσ (G) · e, σ), where ρσ (G) · e is a symmetric space defined by a symmetric pair
(G, F (σ, G)), which is realized by the imbedding G/F (σ, G) → G ; gF (σ, G) 7→ gσ(g)−1 .
(Makiko Sumi Tanaka) Department of Mathematics, Faculty of Science and Technology, Tokyo University of Science, Noda, Chiba, 278-8510, Japan
E-mail address: tanaka makiko@ma.noda.tus.ac.jp

Submanifolds of Symmetric Spaces and Their Time Evolutions

Antipodal sets of compact
symmetric spaces and polars ofcompact Lie groups
Makiko Sumi Tanaka
Tokyo University of Science

Submanifolds of Symmetric Spaces andTheir Time Evolutions, March 5-6, 2021
Online（Zoom）

Joint work with Hiroyuki Tasaki

Contents
1. Introduction
2. Relations between antipodal sets and polars
3. Polars of disconnected compact Lie groups
4. Examples

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1. Introduction
M : a Riemannian manifold
M is called a Riemannian symmetric space
if for ∀x ∈ M , the point symmetry sx at x
is given, i.e., (i) sx is an isometry of M ,(ii) sx ◦ sx = idM , (iii) x is an isolated fixed
point of sx.
• The diﬀerential (dsx)x is −idTxM .
• When M is connected, sx is uniquely determined by (i)-(iii) and sx is the geodesic
symmetry.

F (sx, M ) := {y ∈ M |sx(y) = y}A connected component of F (sx, M ) is calleda polar w.r.t. x.
By (iii), {x} is a polar w.r.t. x, called thetrivial polar.
• A polar M + of positive dimension is a totally geodesic submanifold and hence M + isa Riemannian symmetric space. The point
symmetry at y ∈ M + is given by sy |M + .
• Rn: Euclidean space, F (sx, Rn) = {x}• S n: a sphere, F (sx, S n) = {x, −x}Submanifolds of Symmetric Spaces and Their Time Evolutions• P n: the projective space, F (sx, P n) = {x} ∪P n−1(∵) Set K = R, C, or H and denote P n by KP n.
Since sx is induced by the reflection along
x in Kn+1, F (sx, KP n) ={x} ∪ {1−dim. subspaces in x⊥}(= KP n−1).
• If M is of noncompact type, F (sx, M ) ={x}.
Hereafter we consider the case where M iscompact.

• A compact connected Riem. sym. sp. M is(i) of compact type (I(M ) is compact andsemisimple), (ii) a torus, or a product of (i)and (ii) locally.
A: a subset of M
A is called an antipodal set if for ∀x, y ∈ A,
sx(y) = y holds.
For ∀x ∈ A, A ⊂ F (sx, M ). x is an isolated
point in F (sx, M ) as well as in A. Thus A is
discrete. Hence an antipodal set is finite.

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The 2-number of M is #2M := max{|A| | A ⊂M : an antipodal set}.
If A satisfies |A| = #2M , A is called great.
If A ⊂ A′ implies A = A′, we say A is maximal.
• A great antipodal set is maximal but theconverse is not true.
• #2S n = 2 and {x, −x} is a great antipodal
set.
Bang-Yen Chen and Tadashi Nagano gave
detailed studies of the 2-numbers (ChenNagano, 1988).

In the past ten years there was progressin the research of antipodal sets. Our interest is in maximal antipodal sets themselves rather than their cardinalities.

We

are working on the classification of maximal antipodal sets.
• In (T.-Tasaki, 2017) we classified max. antip.
subgr. of some classical cpt. Lie groups G.
• In (T.-Tasaki, 2020) we classified max. antip.
sets of some classical cpt. Riem. sym. sp. M .
The basic principle is to make use of anSubmanifolds of Symmetric Spaces and Their Time Evolutions

embedding of M into G as a polar w.r.t.
the identity element and apply the classification of max. antip. subgr. of G.
• In order to continue the classification ofmax. antip. sets for some other classical cpt.
Riem. sym. sp. M , we need a realization of
M as a polar of a disconnected cpt. Lie gr.
• Chen-Nagano and Nagano gave detailed
studies of polars of connected cpt. Riem.
symmetric spaces.
• We studied polars of disconnected cpt. Liegroups (T.-Tasaki, submitted).

2. Relations between antipodal sets and polars
G: a compact Lie groupe: the identity element of G
G0: the identity component of G
∃ a biinvariant Riemannian metric on G
G is a compact Riem. symmetric space.
∀x ∈ G, sx(y) = xy −1x (y ∈ G)
• se(y) = y −1, sx(y) = Lx ◦ se ◦ Lx−1 (y)F (se, G) = {x ∈ G | x2 = e}

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F (se, G) =

r
∪
j=0

+
+
G+
j , Gj : a polar, G0 = {e}In general, when a polar consists of a single
point x, we call x a pole.
Proposition 1
ZG(G0): the centralizer of G0 in G
Z̃2(G) := ZG(G0) ∩ F (se, G)• The set of poles coincides with Z̃2(G).
+• For a point x in G+
j , Gj = {Ig (x) | g ∈ G0},

where Ig (x) = gxg −1.

Hence each polar is a G0-conjugacy classof involutive elements.
A: an antipodal set of G
We can assume e ∈ A by left (or right) translations. Then,• x2 = e (x ∈ A), xy = yx (x, y ∈ A).

∼• If A is maximal, A is a subgroup =Z2 × · · · × Z2.
We call such A a maximal antipodal subgroup.

Submanifolds of Symmetric Spaces and Their Time Evolutions

Example. G = O(n): the orthogonal group
G0 = SO(n)1n: the identity matrix
Ij = daig(−1,. {z. . , −1}, 1, . . . , 1) ∈ O(n)|j+Gj = {gIj g −1 | g ∈ SO(n)}∼ SO(n)/S(O(j) × O(n − j))== Gj (Rn): the real Grassmann mfd.
A0 = {diag(ϵ1, . . . , ϵn) | ϵi = ±1} is a maximal
antipodal subgroup of O(n).
Z̃2(O(n)) = {±1n}• A0 is a unique max. antip. subgr. of O(n)up to conjugation, while a max. antip. subgr.
of O(n)/{±1n} is not unique up to conjugation when n is even and n ≥ 4.
M = G+
j : a polar of positive dim.
M is a connected cpt. Riem. sym. sp.
x0 ∈ M , M = {Ig (x0) | g ∈ G0}• I0(M ) = {Ig |M | g ∈ G0}• If A is an antip. set of M , then A ∪ {e} isan antip. set of G.

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OCAMI Reports Vol. 2 (2021)• ∃Ã: a max. antip. subgr. A ∪ {e} ⊂ Ã
• If A is maximal in M , then A = M ∩ Ã.
C1, . . . , Ck : G0-conjugacy classes of maxi.
antip. subgr. of G
Bs: a representative of Cs (1 ≤ s ≤ k)
(We gave their explicit descriptions for someclassical G.)∃g ∈ G0, 1 ≤ ∃s ≤ k,Ã = Ig (Bs)

A = M ∩ Ã = M ∩ Ig (Bs) = Ig (M ∩ Bs)
Hence A is I0(M )-congruent to M ∩ Bs.

Therefore, a representative of an I0(M )congruence class of maximal antipodal setsof M is one of M ∩ B1, . . . , M ∩ Bk .
• Using this principle, for some classical cpt.
Riem. sym. sp. M , we determined I0(M )-cong.
classes of max. antip. sets of M and gave
explicit descriptions of their representatives.
• ∃M , realized as a polar not of a connected
G but of a disconnected G.
e.g., U (n)/O(n), U (2n)/Sp(n)Submanifolds of Symmetric Spaces and Their Time Evolutions

3. Polars of disconnected compact Lie groups
G: a compact Lie group
G0: the identity component of G
G = G0 ∪∪λ∈Λ

Gλ, Gλ: a conn. component

F (se, G) = (F (se, G) ∩ G0) ∪∪λ∈Λ(F (se, G) ∩ Gλ)

We know F (se, G) ∩ G0 by Chen-Nagano.
We study F (se, G) ∩ Gλ.
If F (se, G) ∩ Gλ ̸= ∅, for ∀xλ ∈ Gλ ∩ F (se, G) wehave Gλ = G0xλ = xλG0.

Ixλ (Ixλ (y) = xλyx−1λ ) is an involutive automorphism of G0.

The action defined by g.h = ghIxλ (g)−1 (g, h ∈G0) is called the twisted conjugate action
by Ixλ . (It is a Hermann action.)Tλ: a maximal torus of the identity comp. ofF (Ixλ , G0).
By a property of Hermann actions we have:
Proposition 2 Gλ =

∪

g∈G0

(It is well-known G0 =g(xλTλ)g −1

∪

g∈G0

imal torus T of G0.)

gT g −1 for a max-

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F (se, G) ∩ Gλ =

∪

g∈G0

g{x ∈ xλTλ | x2 = e}g −1

In order to determine F (se, G) ∩ Gλ, it isenough to determine {x ∈ xλTλ | x2 = e} andG0-conjugacy classes of each element of
the set.
We can carry out them for each G on acase-by-case argument.
On the other hand, we have the following:
Proposition 3 Assume Gλ ∩ F (se, G) ̸= ∅.
(1) G0 ∪ Gλ is a subgroup.

(2) For xλ ∈ Gλ ∩ F (se, G), G0 ∪ Gλ is isomorphic to G0 ⋊⟨Ixλ ⟩, where ⟨Ixλ ⟩ is the subgroupof Aut(G0) generated by Ixλ .
Hence, the determination of polars of G is
reduced to the determination of polars of
G0 ⋊ ⟨Ixλ ⟩.
G0 ⋊ ⟨Ixλ ⟩ consists of two connected components:
G0 ⋊ ⟨Ixλ ⟩ = {(g, id) | g ∈ G0} ∪ {(g, Ixλ ) | g ∈ G0}

Submanifolds of Symmetric Spaces and Their Time Evolutions

The group operation of G0 ⋊ ⟨Ixλ ⟩:For g, h ∈ G0, e′ := id, τ := Ixλ ,(g, e′)(h, e′) = (gh, e′), (g, e′)(h, τ ) = (gh, τ ),(g, τ )(h, e′) = (gτ (h), τ ), (g, τ )(h, τ ) = (gτ (h), e′).
Proof of Prop. 3: (1) is easily seen by thegroup operation. (2) φ : G0 ⋊ ⟨Ixλ ⟩ → G0 ∪ Gλdefined by φ(g, id) = g, φ(g, Ixλ ) = gxλ gives a
Lie group isomorphism.
G: a connected cpt. Lie group
σ: an involutive atumorphism of Gê = (e, id): the identity element of G ⋊ ⟨σ⟩Theorem 4
F (sê, G ⋊ ⟨σ⟩) = (F (se, G), id) ∪ (F (se ◦ σ, G), σ)In particular, each connected component of(F (se ◦σ, G), σ) is a polar of G⋊⟨σ⟩. Moreover,the conn. comp. of (F (se ◦σ, G), σ) containing(e, σ) coincides with (ρσ (G) · e, σ), where ρσis the twisted conjugate action by σ, and∼ G/F (σ, G).
ρ (G) · e =σProof of Thm. 4：
F (sê, G ⋊ ⟨σ⟩) = F (sê, (G, id)) ∪ F (sê, (G, σ))

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F (sê, (G, id)) = (F (se, G), id)
F (sê, (G, σ)) = (F (se ◦ σ, G), σ)(∵) ∀g ∈ G,sê(g, σ) = (g, σ)⇔ (g, σ) = (g, σ)−1 = (σ(g −1), σ)⇔ g = σ(g −1)⇔ se ◦ σ(g) = g
As stated before, if we obtain the classification of max.antip. sugr. of G ⋊ ⟨σ⟩, we candetermine max. antip. sets of G/F (σ, G).

4. Examples
U (n): the unitary group
F (s1n , U (n)) ={x ∈ U (n) | x2 = 1n} =

n
∪
j=0

{g Ij g −1 | g ∈ U (n)}Ij = diag(−1,. {z. . , −1}, 1, . . . , 1) ∈ U (n)|jThe polars of U (n) w.r.t. 1n is:{1n}, {−1n},
U (n)/(U (j) × U (n − j)) = Gj (Cn)(1 ≤ j ≤n − 1) the complex Grassmann mfd.
τ (g) := ḡ (g ∈ U (n))

Submanifolds of Symmetric Spaces and Their Time Evolutions

171τ is an involutive autom. of U (n)G = U (n) ⋊ ⟨τ ⟩, ⟨τ ⟩ = {e′, τ }G = {(g, e′) | g ∈ U (n)} ∪ {(g, τ ) | g ∈ U (n)} · · · (∗)
We write (g, e′) by g, and (g, τ ) by gτ .
(∗) ⇝ G = U (n) ∪ U (n)τF (sê, G) = (F (sê, G) ∩ U (n)) ∪ (F (sê, G) ∩ U (n)τ )F (sê, G) ∩ U (n) = F (s1n , U (n)) =

n
∪

j=0

Gj (Cn)We study F (sê, G) ∩ U (n)τ by using Thm. 4.
T : a maximal torus of F (τ, U (n)) = O(n)U (n)τ =∪g(τ T )g −1 (by Prop. 2)g∈U (n)F (sê, G)∩U (n)τ =∪g{x ∈ τ T | x2 = 1n}g −1

g∈U (n)So we study {x ∈ τ T | x2 = 1n}. We can takeT ⊂ O(n)
as




R(θ1)



T =
























...
R(θk )

cos θ − sin θ,R(θ) = sin θ cos θ

(1)












θ1, . . . , θk ∈ R,

k = ⌊n2⌋∀t ∈ T, τ t = (1n, τ )(t, e′) = (τ (t), τ ) = tτ,(τ t)2 = τ 2t2 = t2





















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Hence,{x ∈ τ T | x2 = 1n} = τ {t ∈ T | t2 =1n}







ϵ112
















.



.
.



 ϵ1, . . . , ϵ = ±1 .
=τ
k









ϵk 12


















(1)
F (sê, G) ∩ U (n)τ =∪gτ {t ∈ T | t2 = 1n}g −1

g∈U (n)• ∀t ∈ T, ∀g ∈ U (n), g(τ t)g −1 = g t tg τ
• Since (i12)(−12)(i12) = 12,∀t ∈ T, t2 = 1n, ∃h ∈ U (n) s.t. h t th = 1n.
Hence, if t ∈ T, t2 = 1n, {g(τ t)g −1 | g ∈ U (n)} ={g t tg | g ∈ U (n)}τ = {g 1n tg | g ∈ U (n)}τ.

So F (sê, G) ∩ U (n)τ = {g 1n tg | g ∈ U (n)}τ
Here g1n tg = g1nḡ −1 = g1nτ (g)−1 = ρτ (g)(1n).
ρτ : the twisted conjugate action by τ .
Hence {g 1n tg | g ∈ U (n)} is an orbit of ρτ (G)through 1n.
g 1n tg = 1n ⇔ tg = g −1 = tḡ ⇔ g ∈ O(n)∼ U (n)/O(n) (connected)
F (s , G) ∩ U (n)τ =ê

U (n)/O(n) is realized as a polar of U (n)⋊⟨τ ⟩.
(U (n)/O(n) is not realized as a polar of aconnected compact Lie group.)

Submanifolds of Symmetric Spaces and Their Time Evolutions

Thank you for your kind attention.

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