The 11th OCAMI-KNU RIRCM Joint Diﬀerential Geometry Workshop on
Submanifolds and Lie Theory
March 20 (Sun.)-March 23 (Wed.), 2016.

PROGRAM & ABSTRACT (provisional!)
March 20 (Sunday)
9:30-

Open and Tea

10:00-11:00

Young Jin Suh（Kyungpook National University, Korea)“ Recent progress on complex quadrics in Hermitian Symmetric Spaces ”
Abstract: In this talk we will give some background on the geometry of complex quadric
Qm = SO(m + 2)/SO(2)xSO(m) which can be regarded as Hermitian Symmetric Spacewith rank 2 of compact type. We will give some detailed explanations about the progress onreal hypersurfaces in the complex quadric Qm for Ricci parallel, harmonic curvature, normal
Jacobi operator, pseudo-Einstein real hypersurfaces, and pseudo-anti commuting and Ricci
soliton problem etc.

11:10-12:10

Jost-Hinrich Eschenburg“ Submanifolds and Bott Periodicity (Joint work with Bernhard Hanke) ”
Abstract: Some compact symmetric spaces have poles and equators like spheres. Poles
are points with the same geodesic symmetry, and equators contain the midpoints of shortest
geodesics between poles. These are reﬂective submanifolds called centrioles. Some centrioles
have topological meaning as homotopy approximations of the full path space, thus iterated
centrioles serve to compute higher homotopy groups. This is the basic idea of Milnor’s
version of Bott’s periodicity theorem. We want to show that this idea can be applied also tothe vector bundle versions of the periodicity theorem. As an example, we will talk about anew proof for the theorem of Atiyah, Bott, and Shapiro (1964) which relates vector bundles
over spheres to Cliﬀord modules; these in turn are related to iterated centrioles of theclassical groups. (Common work with Bernhard Hanke, Augsburg)1

13:30-14:30

Naoyuki Koike（Tokyo University of Science, Japan)“ Volume-preserving mean curvature ﬂow for tubes in symmetric spaces ”
Abstract: In this talk, we ﬁrst state the evolutions of the radius function and its gradientalong the volume-preserving mean curvature ﬂow starting from a tube (of nonconstant radius) over a compact closed domain of a reﬂective submanifold in a symmetric space undercertain condition for the radius function. Next, we prove that the tubeness is preserved
along the ﬂow in the case where the ambient space is a rank one symmetric space of compact type (other than a sphere), the reﬂective submanifold is an invariant submanifold andthe initial radius function is radial. Furthermore, in this case, we prove that the ﬂow exists
in inﬁnite time or reaches to one of the invariant submanifold, a focal submanifold of theinvariant submanifold and the cut locus of the intersection point in a normal umbrella ofthe invariant submanifold.

14:40-15:40

Marcos Alexandrino （Institute of Mathematics and Statistics, University of Sao
Paulo, Brasil) (1)“ Mean curvature ﬂow of orbits of isometric actions ”
Abstract: Given a proper isometric action on a compact manifold, we study the meancurvature ﬂow equation with a principal orbit of this action as initial datum. We prove thatany ﬁnite time singularity is a singular orbit, and the singularity is of type I. These results
are proved in the more general context of Singular Riemannian foliations and generalize
previous results of Liu and Terng, Pacini and Koike. This talk is based on a joint workwith Dr. Marco Radeschi (wwu- Munster) and is aimed at a broad audience of students,faculties and researchers. M.M. Alexandrino and M. Radeschi, Mean curvature ﬂow of
singular Riemannian foliations, to appear in The Journal of Geometric Analysis (2015) see
also arXiv:1408.543

15:50-16:50

Takayuki Okuda（Hiroshima University, Japan)“ Pairs of conjugacy classes of reﬂective submanifolds of Riemmanian symmetricspaces with discrete intersections ”
Abstract: Let M = G/K be a Riemannian symmetric space of compact or non-compact
type. In this talk, we classify pairs of reﬂective submanifolds L1 and L2 in M such that theintersection of L1 and gL2 is discrete in M for any g in G. We will also talk about relations between such pairs (L1 , L2 ) and discontinuous groups for certain pseudo-Riemmaniansymmeric spaces.

17:00-18:00

Juergen Berndt（King’s College London, UK)“Compact homogeneous Riemannian manifolds with low co-index of symmetry”
Abstract: The co-index of symmetry measures in how far a homogeneous Riemannian
manifold deviates from being a Riemannian symmetric space. In the talk we will discuss
some general structure theory for compact homogeneous Riemannian manifolds in relation
to the co-index of symmetry. We also present a classiﬁcation of compact homogeneous
Riemannian manifolds with co-index of symmetry less or equal than three. This is joint
work with Carlos Olmos (Cordoba) and Silvio Reggiani (Rosario), to appear in Journal of
the European Mathematical Society (see also arXiv:1312.6097).

19:00-

Party (at a restaurant near the Abiko station).

March 21 (Monday)
9:30-10:30

Jong Ryul Kim（Kunsan National University)“A gradient Ricci soliton with a regular hypersurface of a potential function f andHessf (∇f, ∇f ) = 0”
Abstract: For a complete oriented Riemannian manifold M of a gradient Ricci soliton
with a regular level hypersurface Hs = {x ∈ M |f (x) = s} for a potential function f∇fwith a unit normal vector ﬁeld N = |∇f| on Hs , we show that if Ric(N, N ) = λ and⊥DEi R = 2Ric(∇f, Ei ) is constant for a local geodesic frame {Ei }n−1i=1 of (∇f ) , then M is
Ricci ﬂat and each level hypersurface is totally geodesic. Moreover M is a product manifold
Hs × R.

10:40-11:40

Hiroshi Tamaru（Hiroshima University, Japan)“Realizations of some contact metric manifolds as Ricci soliton real hypersurfaces”
Abstract: Ricci soliton contact metric manifolds with some nullity conditions have recently been studied by Ghosh and Sharma. Whereas the gradient case is well-understood,
they provided a list of candidates for the non-gradient case. These candidates can be realized as Lie groups, but one only knows the bracket relations of the Lie algebras, which are
hard to be understood apart from the three-dimensional case. In this talk, we will study
these spaces with higher-dimensions, and prove that the simply-connected ones can be realized as homogeneous real hypersurfaces in noncompact two-plane Grassmannians. These
realizations enable us to prove, in a Lie-theoretic way, that all of them are actually Riccisolitons. This talk is based on a joint work with Jong Taek Cho, Takahiro Hashinaga, Akira
Kubo, and Yuichiro Taketomi.

13:00-13:30

Shinji Ohno（Tokyo Metropolitan University, Japan)“ A construction of weakly reﬂective submanifolds in compact symmetric spaces ”
Abstract: A submanifold of a Riemannian manifold is called a weakly reﬂective submanifold if for any point in the submanifold and each normal vector at the point, there
exists a weakly reﬂection which is an isometry on the ambient Riemannian manifold. The
notion of weakly reﬂective submanifold is a generalization of the notion of reﬂective submanifold. Reﬂective submanifolds in irreducible compact symmetric spaces are classiﬁed.
In contrast, examples of weakly reﬂective submanifolds in compact symmetric spaces are
not well known except for examples in hyperspheres. In this talk we will give examples of
weakly reﬂective submanifolds in compact symmetric spaces as orbits of Hermann actions
which are generalizations of isotropy actions of compact symmetric spaces. To construct
weakly reﬂective submanifolds in compact symmetric spaces, we use symmetric triads which
are generalizations of restricted root systems.

15:35-16:05

Hyunjin Lee and Young Jin Suh (Kyungpook National University, Korea)“Some parallel Hopf hypersurfaces in complex hyperbolic two-plane Grassmannians”
Abstract: In this talk, we ﬁrst consider a concept named the cyclic parallelism for theshape operator of a real hypersurface M in complex hyperbolic two-plane Grassmannian.
Moreover, we prove that the cyclic parallelism is to be a Reeb parallelism on M . Hence, we
study the Reeb parallelism by the Levi-Civita and generalized Tanaka-Webster connections
on M . Related to these notions, we give some characterizations of Hopf hypersurfaces incomplex hyperbolic two-plane Grassmannians.

14:10-14:40

Eunmi Pak and Young Jin Suh (Kyungpook National University, Korea)“Results on Jacobi operators of real hypersurfaces in complex two-plane Grassmannians ”
Abstract: In relation to the generalized Tanaka-Webster connection, we consider a new
notion of parallel Jacobi operator for real hypersurfaces in complex two-plane Grassmannians and show results about real hypersurfaces in complex two-plane Grassmannians withgeneralized Tanaka-Webster parallel structure Jacobi operator and normal Jacobi operator.

14:10-14:40

Eunmi Pak and Young Jin Suh (Kyungpook National University, Korea)“Results on Jacobi operators of real hypersurfaces in complex two-plane Grassmannians ”
Abstract: In relation to the generalized Tanaka-Webster connection, we consider a new
notion of parallel Jacobi operator for real hypersurfaces in complex two-plane Grassmannians and show results about real hypersurfaces in complex two-plane Grassmannians withgeneralized Tanaka-Webster parallel structure Jacobi operator and normal Jacobi operator.

14:45-15:15

Changhwa Woo, Hyunjin Lee and Young Jin Suh (Kyungpook National University,Korea)“Real hypersurfaces in complex hyperbolic two-plane Grassmannians with commuting restricted structure Jacobi operators ”
Abstract: In this paper, we introduce a new commuting condition between the structure
Jacobi operator and symmetric (1,1)-type tensor ﬁeld T , that is, Rξ ϕT = T Rξ ϕ, where
T = A or T = S for Hopf hypersurfaces in complex hyperbolic two-plane Grassmannians. By
using simultaneous diagonalzation for commuting symmetric operators, we give a complete
classiﬁcation of real hypersurfaces in complex hyperbolic two-plane Grassmannians with
commuting condition respectively.

15:20-15:50

Doo Hyun Hwang and Young Jin Suh (Kyungpook National University, Korea)“Hopf hypersurfaces in complex two plane Grassmannians satisfying recurrent Riccitensors”
Abstract: In this paper, we have introduced a new notion of generalized Tanaka?Webster
Reeb recurrent Ricci tensor of real hypersurfaces in complex two-plane Grassmannians.
Next, we show a non-existence property for real hypersurfaces M in complex two-plane
Grassmannians with such a curvature condition.

15:55-16:25

Gyu Jong Kim and Young Jin Suh (Kyungpook National University, Korea)“Real hypersurfaces in complex two-plane Grassmannians with Killing structure
Jacobi operator”
Abstract: In this talk, we introduce a notion of the complex two-plane Grassmannians
and a geometric meaning of Killing structure Jacobi operator. Then we give a non-existence
theorem for a real hypersurface in complex two-plane Grassmannians satisﬁes the conditionof Killing structure Jacobi operator.

17:00-17:50

Jost-Hinrich Eschenburg （University of Augsburg, Germany)“What is a Penrose tiling? (Special Talk)”
Abstract: As ﬁnite beings with only a ﬁnite set of ideas in mind, how can we invent
tilings covering the whole inﬁnite plane (complete tilings)? In art, two answers have beendeveloped: Periodicity (a ﬁnite domain repeats itself in two directions) and Self Similarity(a small area determines the pattern for a larger area). In 1974, Roger Penrose introduced
a class of tilings which are not periodic but have a certain self similarity property. They are
constructed from two types of tiles, based on the geometry of the regular pentagon. So far
there is no valid constuctive deﬁnition for Penrose tilings which are extendible to complete
ones. Matching rules (see Wikipedia article on Penrose tilings) are not suﬃcient as we willsee. One way to construct complete tilings is by projecting a strip of a higher dimensional
regular grid. However, we will give an example of a Penrose tiling which does not arise in thisway. We use an extremely narrow deﬁnition of Penrose tilings which seems to completely
determine those patterns, but still it allows uncountably many diﬀerent patterns, including
all projection tilings. This ﬁnding was heavily inﬂuenced by some 400 years old Islamic
patterns at Isfahan, Iran. (Common work with H.J. Rivertz, Trondheim, Norway)

March 22 (Tuesday)
10:00-11:00

Hiroyuki Tasaki（University of Tsukuba, Japan)“Maximal antipodal sets of oriented real Grassmann manifoldss”
Abstract: An antipodal set in a compact Riemannian symmetric space is a subset where
the restriction of the symmetry at each point is the identity, which was introduced by Chenand Nagano. A maximal antipodal set is a kind of frame of a compact Riemannian symmetric space. In this talk I mainly treat maximal antipodal sets of oriented real Grassmann
manifolds. We can reduce the classiﬁcation of these to a combinatorial problem and classify
these in the case where the rank is less than ﬁve. Moreover we can estimate the cardinalities
of maximal antipodal sets in cases of higher rank.

11:10-12:10

Makiko Sumi Tanaka（Tokyo University of Science)“Antipodal sets of compact Riemannian symmetric spaces ”
Abstract: The isotropy actions of compact symmetric spaces are typical examples ofHermann actions. Hermann actions have nice properties. For example these are hyperpolar
actions and also variational complete actions. We study the orbits of Hermann actions. Inorder to do this, we introduce a notion of a symmetric triad, which is a generalization of anirreducible root system.

13:30-14:30

Leonardo Biliotti （University of Parma, Italy)“Stability of measures on Kahler manifolds ”
Abstract: Let M be a Kahler manifold and let K be a compact group that acts on M in a
Hamiltonian fashion. We will study the action of K and its complexiﬁcation G on the spaceof probability measures on M. First of all we identify an abstract setting for the momentum
mapping and give numerical criteria for stability, semi-stability and polystability. Next weapply this setting to the action of G on measures. We get various stability criteria formeasures on Kahler manifolds. The same circle of ideas gives a very general surjectivity
result for a map originaly studied by Hersch and Bourguignon-Li-Yau. This is a joint workwith Dott. Alessandro Ghigi.

14:40-15:40

Marcos Alexandrino （Institute of Mathematics and Statistics, University of Sao
Paulo, Brasil) (2)“Isometry ﬂows on orbit spaces”
Abstract: In this talk, we discuss the following result: Given a proper isometric action
K × M → M on a complete Riemannian manifold M then each continuous isometric ﬂowon the orbit space M/K is smooth, i.e., it is the projection of an K-equivariant smooth
ﬂow on the manifold M . The ﬁrst application of our result concerns Molino’s conjecture,
which states that the partition of a Riemannian manifold into the closures of the leaves ofa singular Riemannian foliation is still a singular Riemannian foliation. We prove Molino’sconjecture for the main class of foliations considered in his book, namely orbit-like foliations.
We also discuss smoothness of isometric actions on orbit spaces. This talk is based on ajoint work with Dr. Marco Radeschi (wwu- Munster) and is aimed at a broad audience ofstudents, faculties and researchers in Geometry. ps: I will also brieﬂy explain current works
and how this talk is related to holonomy foliation that is associated to every connections onﬁber bundles.

15:50-16:50···“ ··· ”

Abstract: · · ·March 23 (Wednesday)
10:00-11:00

Takashi Sakai (Tokyo Metropolitan University, Japan)“Biharmonic homogeneous submanifolds in compact symmetric spaces”
Abstract: In 1983, J. Eells and L. Lemaire extended the notion of harmonic map between Riemannian manifolds to that of biharmonic map, which is deﬁned as a critical pointof the bienergy functional. G.Y. Jiang studied the ﬁrst and second variation formulas of
the bienergy functional and obtained the Euler-Lagrange equation, which is a fourth order
PDE. In this talk, ﬁrst we study biharmonic submanifolds in Einstein manifolds, and givea necessary and suﬃcient condition for a submanifold whose tension ﬁeld is parallel to be
biharmonic. For orbits of commutative Hermann actions in compact symmetric spaces, thiscondition can be described in terms of symmetric triads. By using this criterion, we construct some proper biharmonic submanifolds in compact symmetric spaces, which are orbitsof commutative Hermann actions. Here, proper biharmonic means that biharmonic, but notharmonic. This talk is based on a joint work with Shinji Ohno and Hajime Urakawa.

11:10-12:10

Yoshihiro Ohnita（Osaka City University & OCAMI)“Hamiltonian non-displaceability of Gauss images of isoparametric hypersurfaces(Joint work with Hiroshi Iriyeh, Hui Ma and Reiko Miyaoka)”
Abstract: In this talk we discuss the Hamiltonian non-displaceability of Gauss images of
isoparametric hypersurfaces in the spheres as Lagrangian submanifolds embedded in complex
hyperquadrics.