INTRODUCTION TO INTEGRAL GEOMETRYIN RIEMANNIAN HOMOGENEOUS SPACES
TAKASHI SAKAI
Abstract. In these talks, we shall study fundamentals of integral geometry in Riemannian homogeneous spaces, especially kinematic formulae and their applications.

1. Introduction
In the last century, the branch of integral geometry was initiated from the idea ofgeometric probability by Blaschke, Chern, Santaló and other geometers. (Santaló’s book[16] is a good reference for the history and introduction to integral geometry.) Let X bea set of certain geometric objects, such as points, lines, motions and so on. When we
consider a geometric quantity for each element of X, it can be regarded as a function
on X. We shall give a measure on X, and evaluate the integral of that function on
X as the “average” of that geometric quantity. This integration sometimes leads togeometric consequences. One of the most important problem to do this is how to deﬁne
an appropriate measure on X. When X is a homogeneous space, a method of Lie groups
provides invariant measures on X. In this series of talks we shall study such geometricintegral formulae in Riemannian homogeneous spaces.
Let G/K be a homogeneous space with an invariant Riemannian metric. Consider two
compact submanifolds M and N of G/K, one ﬁxed and the other moving under the actionof g ∈ G. Then it becomes a measurable function on G if we assign an integral invariantI(M ∩ gN ) of submanifold M ∩ gN for each g ∈ G. It is called the kinematic formulathat represents the equality between the integral∫I(M ∩ gN )dµG (g)

(1.1)
G

and some geometric invariants of M and N , where dµG is the invariant measure of G. In
particular, it is called Poincaré formula if we take vol(M ∩ gN ) as an integral invariant in(1.1). Furthermore, Chern [3] and Federer [4] obtained a kinematic formula for integral
invariants which appear in coeﬃcients of the Weyl tube formula and the generalized GaussBonnet formula. In 1993 Howard [5] deﬁned integral invariants induced from an invariant
homogeneous polynomial of the second fundamental form of M ∩ gN , and he achieved
more general kinematic formula. He put the Poincaré formula, the Chern-Federer formulaetc. into a uniform shape.
The kinematic formulae are interesting in their own forms, moreover they have been
applied to several geometric problems. For example, Zhou [20] was obtained an extensionof Chen’s formula, and applied it to describe a suﬃcient condition for Hadwiger’s containment problem in higher dimensions. The Poincaré formula has been applied to show
some volume minimizing properties of certain submanifolds (see [6], [8], [9] and [18]).
12. Integral geometry in the plane
We begin with classical geometric integral formulae in the plane. One of the mostfundamental ones is the kinematic formula of intersection numbers of two curves in theplane.
Let us consider the set I(R2 ) of all isometries of R2 which preserve the orientation. An
element of I(R2 ) can be expressed as[ ][][ ] [ ]xcos ϕ − sin ϕxu
T (ϕ, u, v) :
7−→
+
.
y
sin ϕ cos ϕyv
A triad (ϕ, u, v) of real numbers provides a local coordinate and the diﬀerentiable manifold
structure on I(R2 ). As a Lie group, I(R2 ) is isomorphic to the semi-direct productSO(2) n R2 . We give a Riemannian metric on I(R2 ) by the product metric of SO(2) andR2 . Then it is known the following theorem.
Theorem 2.1 (Poincaré formula). For two curves c0 and c1 in R2∫#(c0 ∩ gc1 )dµ(g) = 4 Length(c0 )Length(c1 )I(R2 )

holds. Here dµ denotes the Riemannian measure on I(R2 ).
From the Poincaré formula, we can obtain some other kinematic formulae. Let S(x, r)
denote a circle of radius r centered at x in R2 . For a ﬁxed r, the set of all circles of radiusr in R2 is identiﬁed with R2 .
Theorem 2.2. For a curve in R2∫#(c ∩ S(x, r))dµ(x) = 4r Length(c)R2holds.
Proof. Apply the Poincaré formula for a circle c1 = S(O, r) centered at the origin O.
Then we have∫
4 · 2πr · Length(c) =#(c ∩ gS(O, r))dµ(g)I(R2 )∫∫ 2π#(c ∩ T (ϕ, u, v) · S(O, r))dϕ dudv

=
R2

∫

0

= 2π
R2

#(c ∩ S((u, v), r))dudv,

because S(O, r) is invariant under the rotation.



Next we shall consider the set L(R2 ) of all lines in R2 . An element of L(R2 ) can be
expressed by a pair (r, θ) of real numbers asl(r, θ) = {(x, y) ∈ R2 | x cos θ + y sin θ = r}.
Then, (r, θ) provides a local coordinate and the diﬀerentiable manifold structure on L(R2 ).
Actually L(R2 ) is diﬀeomorphic to Möbius strip, therefore is non-orientable. We give a
Riemannian metric g on L(R2 ) which is locally isometric to R2 , that is, g = dr2 + dθ2 .
Then similarly with Theorem 2.2 we can obtain the following theorem from the Poincaré
formula, however a line has inﬁnite length.
2

Theorem 2.3 (Crofton formula). For a curve c in R2∫#(c ∩ l)dµ(l) = 2 Length(c)L(R2 )

holds. Here dµ denote the Riemannian measure on L(R2 ).
For a closed curve c in R2 , let dc (θ) denote the width of c with respect to the direction(cos θ, sin θ). Then from the Crofton formula, we get the following.
Theorem 2.4 (Barbier’s theorem). For a closed convex curve c in R2∫ π
dc (θ)dθ = Length(c)0holds. Hence, if c has constant width d, then Length(c) = πd.
Proof. From the Crofton formula,∫∫ π∫ ∞
2 Length(c) =#(c ∩ l)dµ(l) =#(c ∩ l(r, θ))dr dθ.
L(R2 )

0−∞

For a ﬁxed θ, there exists a closed interval [rθ , rθ + dc (θ)] so that l(r, θ) meets c if andonly if r ∈ [rθ , rθ + dc (θ)]. Since c in convex, l(r, θ) has exactly two intersection points
with c for r ∈ (rθ , rθ + dc (θ)). Therefore we have∫ π ∫ rθ +dc (θ)∫ π
2 Length(c) = 2
dr dθ = 2
dc (θ)dθ.
0

rθ

0


We also know the following remarkable kinematic formula.
Theorem 2.5 (Blaschke formula). Let c0 and c1 be simply closed curves in R2 , and let
D0 and D1 be domains encircled by c0 and c1 respectively. We denote by χ(D) the numberof connected components of a domain D in R2 . Then∫
χ(D0 ∩ gD1 )dµ(g) = 2π(Area(D0 ) + Area(D1 )) + Length(c0 )Length(c1 )I(R2 )

holds.
3. Kinematic formulae in Riemannian homogeneous spaces
In this section we shall study the general theory of the kinematic formulae in Riemannian homogeneous spaces due to Howard. Refer to his paper [5] for details.
Let G be a Lie group and K a compact subgroup of G. We assume that G has a left
invariant metric that is also right invariant under K. Then a homogeneous space G/K
has an invariant metric. Throughout this paper, we also assume that G is unimodular,that is | det Ad(g)| = 1 for any g ∈ G. We denote by T = To (G/K) the tangent space of
G/K at the origin o. Let V be a linear subspace of T . A submanifold M of G/K is said tobe of type V if and only if for each x ∈ M there exists a gx ∈ G such that (gx )∗ V = Tx M .
3

3.1. Generalized Poincaré formula. Let E be an n-dimensional real vector space withan inner product ⟨·, ·⟩. For two vector subspaces V and W of dimensions p and q in E with
p + q ≤ n, we take orthonormal bases v1 , . . . , vp and w1 , . . . , wq of V and W respectively.
Then we deﬁne σ(V, W ), the angle between V and W , byσ(V, W ) = ∥v1 ∧ · · · ∧ vp ∧ w1 ∧ · · · ∧ wq ∥,where∥x1 ∧ · · · ∧ xk ∥2 = det(⟨xi , xj ⟩).
This deﬁnition is independent of the choice of orthonormal bases.
Now let us deﬁne the angle between subspaces of tangent spaces of a Riemannian
homogeneous space G/K. For x and y in G/K and vector subspaces V in Tx (G/K) and
W in Ty (G/K) we deﬁne σK (V, W ), the angle between V and W , by∫−1
−1
σK (V, W ) =σ((gx )−1
∗ V, k∗ (gy )∗ W )dµK (k),K

where gx and gy are elements of G so that gx o = x and gy o = y. We note that thisdeﬁnition is independent of the choice of gx and gy .
With this notation, the generalized Poincaré formula for homogeneous spaces can bestated as follows:
Theorem 3.1 (Howard [5]). Let M and N be submanifolds of G/K with dim M +dim N ≥dim(G/K). Then
∫
∫
(3.1)
vol(M ∩ gN )dµG (g) =σK (Tx⊥ M, Ty⊥ N )dµM ×N (x, y)
M ×NGholds.
Equality (3.1) implies that if M is a submanifold of G/K of type V and N of type W
for some subspaces V and W of T , then σK is a constant function on M × N and∫(3.2)
vol(M ∩ gN )dµG (g) = σK (V ⊥ , W ⊥ )vol(M )vol(N ).
G

It is clear that if G/K is a real space form then all (dim V )-dimensional submanifolds areof type V . So (3.2) yields the Poincaré formula in real space forms.
Theorem 3.2. Let G/K be a real space form of dimension n with K ∼= SO(n). Then forsubmanifolds M and N in G/K of dimensions p and q with p + q ≥ n∫
vol(SO(n + 1))vol(S p+q−n )vol(M )vol(N )vol(M ∩ gN )dµG (g) =vol(S p )vol(S q )Gholds.
In the case where G/K is a complex space form, any p-dimensional complex submanifold
is of type V for any p-dimensional complex subspace V in To (G/K), and any Lagrangiansubmanifold is of type V for any Lagrangian subspace V in To (G/K). Thus we have thefollowing theorems.
Theorem 3.3. Let G/K be a complex space form of complex dimension n with K ∼=U (1) × U (n). Then for submanifolds M and N in G/K of complex dimensions p and q
with p + q ≥ n∫
vol(U (n + 1))vol(CP p+q−n )vol(M )vol(N )vol(M ∩ gN )dµG (g) =vol(CP p )vol(CP q )Gholds.
4

Theorem 3.4. Let G/K be a complex space form of complex dimension n with K ∼=U (1) × U (n). Then for Lagrangian submanifolds M and N in G/K∫(n + 1)vol(U (n + 1))
vol(M )vol(N )#(M ∩ gN )dµG (g) =vol(RP n )2G
holds.
3.2. Kinematic formula in Riemannian homogeneous spaces. For a linear subspace V of T , we deﬁne a vector space II(V ) to be{}
II(V ) = h | h : V × V → V ⊥ ; symmetric bilinear ,where V ⊥ is the normal space of V in T . A second fundamental form of a submanifold of
G/K which passes through o and has V as the tangent space at o is an element of II(V ).
Let K(V ) be the stabilizer of V in K, that is, K(V ) = {k ∈ K | k∗ V = V }. The group
K(V ) acts on II(V ) in the following manner:
(
)
(3.3)
(kh)(u, v) = k∗ h(k∗−1 u, k∗−1 v)(u, v ∈ V )for k ∈ K(V ) and h ∈ II(V ). Here we may consider a polynomial P on the vector
space II(V ) which is invariant under K(V ), that is, P(kh) = P(h) for all k ∈ K(V )and h ∈ II(V ). In addition, let M be a submanifold of G/K of type V . For the second
fundamental form hM
x of M at x ∈ M , we deﬁne
−1

gx M
P(hM
).
x ) = P(hoThen we can deﬁne an integral invariant I P (M ) of M from a polynomial P by
∫
P
(3.4)
I (M ) =P(hMx )dµM (x).
M

We also deﬁne a vector space EII(T ) to be
EII(T ) = {h | h : T × T → T ; symmetric bilinear} .
Since K also acts on EII(T ) in the same way with (3.3), we can deﬁne integral invariantsfrom polynomials on EII(T ) invariant under K in the same manner with (3.4).
With these preliminaries, we can now state the kinematic formulae in Riemannian
homogeneous spaces as follows:
Theorem 3.5. ([5] paragraph 4.10) Let V and W be linear subspaces of T with dim(V ) +dim(W ) ≥ dim(T ), and let P be a homogeneous polynomial of degree l on EII(T ) whichis invariant under K such that∫
(3.5)σ(V ⊥ , k∗ W ⊥ )1−l dµK (k) < ∞.
K

Then there exists a ﬁnite set of pairs (Qα , Rα ) such that(1) each Qα is a homogeneous polynomial on II(V ) invariant under K(V ),(2) each Rα is a homogeneous polynomial on II(W ) invariant under K(W ),(3) deg Qα + deg Rα = l for each α,(4) for any compact submanifolds (possibly with boundaries) M of type V and N oftype W in G/K the kinematic formula
∫
∑
(3.6)
I P (M ∩ gN )dµG (g) =I Qα (M )I Rα (N )Gαholds.
5

The inequality (3.5) is a condition for the convergence of the integration. If G/K is areal space form, then the condition (3.5) can be replaced by the manageable inequalityl ≤ dim(M ) + dim(N ) − dim(G/K) + 1.
3.3. Transfer principle. Under the hypothesis of Theorem 3.5 let G′ be a unimodular
Lie group with dim G′ = dim G and K ′ be a compact subgroup of G′ with isomorphismρ : K → K ′ and vol(K) = vol(K ′ ). We assume that there is a linear isometry ψ : T → T ′such that(∀ k ∈ K).

ψ ◦ k∗ = ρ(k)∗ ◦ ψThen ψ induces following isomorphisms of rings:
{
}
{
}
′
polynomials on EII(T )polynomials
on
EII(T
)
∼
=
invariant under Kinvariant under K ′P{polynomials on II(V )invariant under K(V )

P′

7−→
}

Qα , Rα∼={polynomials on II(ψV )invariant under K ′ (ψV )}Q′α , R′α7−→

Here we denote by P ′ , Q′α , R′α the image of P, Qα , Rα respectively under these isomorphisms. With this notation, if a kinematic formula (3.6) holds in G/K, then a kinematicformula∫∑ ′′′I P (M ′ ∩ gN ′ )dµG′ (g) =I Q α (M ′ )I R α (N ′ )G′α′holds for any compact submanifolds M of type ψV and N ′ of type ψW in G′ /K ′ .

4. Some explicit expressions of kinematic formulae
In this section, we shall give some explicit forms of kinematic formulae stated in Theorem 3.5.
Take an orthonormal basis e1 , . . . , en of T such that e1 , . . . , ep is a basis of V andep+1 , . . . , en is a basis of V ⊥ . Then the components of h ∈ II(V) and H ∈ EII(T) arerepresented byhkij = ⟨h(ei , ej ), ek ⟩(1 ≤ i, j ≤ p, p + 1 ≤ k ≤ n)

Hijk = ⟨H(ei , ej ), ek ⟩(1 ≤ i, j, k ≤ n)

The following polynomials W2l are homogeneous polynomials on II(V ) of degree 2l invariant under O(V ) × O(V ⊥ ).

 k1
hi1 i1
hki11i2 . . . hki11i2l
 hk1
hki21i2 . . . hki21i2l  i2 i1
∑


..
..
..
.
.
W2l (h) =
det 
.
.
.
.
.

 kl
kl
kl
1≤i1 ,...,i2l ≤phi2l−1 i1 hi2l−1 i2 . . . hi2l−1 i2l 
p+1≤k1 ,...,kl ≤nhki2ll i1
hki2ll i2 . . . hki2ll i2l6

We deﬁne homogeneous polynomials, also denoted by W2l , by
Hik11i2 . . . Hik11i2l
Hik11i1
 H k1
Hik21i2 . . . Hik21i2l 
i2 i1


∑


..
..
..
.
.
W2l (H) =
det 

.
.
.
.

 kl
kl
kl
1≤i1 ,...,i2l ≤n


H
H
.
.
.
H
i2l−1 i1
i2l−1 i2i2l−1 i2l1≤k1 ,...,kl ≤n
Hik2ll i1
Hik2ll i2 . . . Hik2ll i2l
on EII(T ) of degree 2l invariant under O(T ). In the both cases, W0 = 1 by deﬁnitions.
A second fundamental form h ∈ II(V ) can be extended to H ∈ EII(T) by
H(u, v) = h(P u, P v)

(u, v ∈ T ),where P : T → V is the orthogonal projection. If H ∈ EII(T) is the extension ofh ∈ II(V ), then we haveW2l (h) = W2l (H).
Furthermore these polynomials W2l are characterized as the invariant polynomials which
vanish on the (extended) second fundamental forms with relative rank less than 2l. For
a submanifold M of G/K, we introduce the integral invariants µ2l (M ) deﬁned byµ2l (M ) = I W2l (M ).
For these integral invariants µ2l , we have the following remarkable formula.
Theorem 4.1 (Chern-Federer formula). Let G/K be a real space form of dimension nwith K ∼= SO(n). Assume that 0 ≤ 2l ≤ p+q −n. Then there exist constants c(p, q, n, i, l)determined by indicated parameters so that∫l∑µ2l (M ∩ gN )dµG (g) =c(p, q, n, i, l)µ2i (M )µ2(l−i) (N )G

i=0holds for any compact submanifolds M and N in G/K of dimensions p and q, respectively.
The value of the constants a(p, q, n, i, l) were computed by Chern [3], Nijenhuis [10].
The space of homogeneous polynomials on II(V ) of degree 2 invariant under O(V ) ×
O(V ⊥ ) is spanned by two polynomials()2∑ ∑∑ ( )2Q1 (h) =hkij ,Q2 (h) =hkii ,
ki,j,ki

where 1 ≤ i, j ≤ p, p + 1 ≤ k ≤ n. And if 2 ≤ p ≤ n − 1 these two polynomials are
independent. Geometrically, Q1 (h) is the square of the norm of the second fundamental
form, and Q2 (h) is p2 times the square of the mean curvature. However, it is convenient
for us to take the basis
W2 = Q2 − Q1 ,Up = pQ1 − Q2 .

For these polynomials we have the following:
Theorem 4.2. Let G/K be a real space form of dimension n with K ∼= SO(n). Assume
that 2 ≤ p + q − n. Then there exist constants a(p, q, n) and b(p, q, n) so that∫
I W2 (M ∩ gN )dµG (g) = a(p, q, n)I W2 (M )vol(N ) + a(q, p, n)vol(M )I W2 (N )∫G
I Up+q−n (M ∩ gN )dµG (g) = b(p, q, n)I Up (M )vol(N ) + b(q, p, n)vol(M )I Uq (N )G

7holds for any compact submanifolds M and N in G/K of dimensions p and q, respectively.
The ﬁrst one is entirely the Chern-Federer formula of degree 2. The second one wassuggested by Howard [5]. Finally Kang-Suh-S. [7] gave the explicit forms completely.
The polynomial Up is characterized as the invariant polynomial which vanishes at anumbilic point. The integral invariant
∫
(
)p/2
p/2
Up
I
(M ) =Up (hMdµM (x)x )M

is an conformal invariant, called the Willmore-Chen functional, of p dimensional submanifold M (see [1], [2], [17]).
5. Transferred kinematic formulae in rank one symmetric spaces
By the transfer principle we can obtain some kinematic formulae in rank one symmetricspaces transferring from the case of real space forms.
Let G/K be a rank one symmetric space, that is one of a sphere S n = SO(n+1)/SO(n),a real projective space RP n = SO(n + 1)/S(O(1) × O(n)), a complex projective space
CP n = U (n + 1)/U (1) × U (n), a quaternionic projective space HP n = Sp(n + 1)/Sp(1) ×Sp(n), the Cayley projective plane CayP 2 = F4 /Spin(9), and their non-compact duals.
Then G/K is isotropic, that is, K acts transitively on the sphere in T by the linear
isotropy representation. Therefore, for v ∈ T we denote by K(v) the stabilizer of v in K,then K/K(v) is homothetic to the unit sphere. It is clear that any real hypersurfaces ina rank one symmetric space is submanifolds of type V for any hyperplane V in T .
Theorem 5.1. ([5] paragraph 3.12) Let G/K be a rank one symmetric space of dimension
n. Let M be a submanifold of dimension p and N a real hypersurface in G/K. Then∫
vol(K)vol(S p−1 )vol(S n )vol(M ∩ gN )dµG (g) =vol(M )vol(N )vol(S p )vol(S n−1 )Gholds.
When G/K is a rank one symmetric space except S n , RP n and RH n , K does not
act transitively on Grp (T ) for p ≥ 2. Therefore Theorem 5.1 is outside of Theorem 3.5.
Howard showed this formula in a geometric way like the transfer principle. Here we prove
this in a diﬀerent way using generalized Poincaré formula.
Proof. Let V ∈ Grp−1 (T ) and v ∈ T . Set W = Rv. Then∫σK (V, W ) =σ(V, k∗ W )dµK (k)K∫= vol(K(v))σ(V, [k]∗ W )dµK/K(v) ([k])
K/K(v)=

vol(K)
vol(S n−1 )∫σ(V, [k]∗ W )dµS n−1 ([k]).
K/K(v)In the second equality the integration on K is reduced to that on K/K(v). The lastequality means that the invariant measure on K/K(v) is normalized to be the unit sphereS n−1 . The last integration on the unit sphere does not depend on V , and is equal for any
rank one symmetric spaces. Therefore we have
σK (V, W ) =vol(K)σ(V, W ).
vol(SO(n)) SO(n)8Form Theorem 3.2 we havevol(SO(n + 1))vol(S p−1 )σSO(n) (V, W ) =.
vol(S p )vol(S n−1 )Thus we obtain the theorem.

Poincaré formula is a kinematic formula for the integral invariant of degree 0, thatis volume functional. A fundamental question is how to express kinematic formulae forother integral invariants in rank one symmetric spaces. From Theorem 4.2 we now havethe following formulae.
Theorem 5.2 ([15]). Let M and N be real hypersurfaces in a rank one symmetric space
G/K of dimension n. Then the following kinematic formulae hold:∫
I W2 (M ∩ gN )dµG (g)
G

=
∫

(
)
vol(K)a(n − 1, n − 1, n) I W2 (M )vol(N ) + vol(M )I W2 (N ) ,vol(SO(n))

I Un−2 (M ∩ gN )dµG (g)

G

=

(
)
vol(K)b(n − 1, n − 1, n) I Un−1 (M )vol(N ) + vol(M )I Un−1 (N ) .
vol(SO(n))
6. Applications of kinematic formulae

6.1. Hadwiger’s containment problem. In 1941, Hadwiger obtained some suﬃcientconditions for one domain to contain another domain in R2 (see [16]). Later Zhou generalized his results to higher dimensions and other real space forms using some famous
kinematic formulae. Here we shall work in the case of R3 for simplicity, although heobtained some results in the more general situations ([20], [21]). The idea of the proof isessentially same.
In this subsection, we use the following notation. For domains Di (i = 0, 1) in R3 , wedenote their volumes by Vi , and the Euler-Poincaré characteristic by χ(·). For surfaces Mior ∂Di (i = 0, 1) in R3 , we denote the surface areas, the total Gaussian curvatures, total(2)mean curvatures and total square mean curvatures by Fi , K̃i , H̃i and H̃i , respectively.
Theorem 6.1 (Chern). Let D0 and D1 be two domains bounded by simple surfaces ∂D0and ∂D1 in R3 . Then∫
χ(D0 ∩ gD1 )dµ(g) = 8π 2 (V0 χ(D1 ) + χ(D0 )V1 ) + 2π(F0 H̃1 + H̃0 F1 )I(R3 )

holds.
Theorem 6.2 (C.S. Chen). Let M0 and M1 be two surfaces in R3 . Then
(∫
)
∫
(2)
(2)
2
κg ds dµ(g) = 2π 3 (3H̃0 − K̃0 )F1 + 2π 3 F0 (3H̃1 − K̃1 )I(R3 )

M0 ∩gM1

holds, where κg denotes the curvature of the intersection curve M0 ∩ gM1 .
9

Theorem 6.3 (Zhou [19]). Let D0 and D1 be connected domains bounded by simple
surfaces ∂D0 and ∂D1 in R3 . Assume that there exists a ﬁnite integer N0 such thatχ(D0 ∩ gD1 ) ≤ N0 for all g ∈ I(R3 ). If
8π(V0 χ(D1 ) + χ(D0 )V1 ) + 2(F0 H̃1 + H̃0 F1 )
(
)) 12
(
(2)
(2)
> 0,
−N0 π F0 F1 (3H̃0 − K̃0 )F1 + F0 (3H̃1 − K̃1 )then there exists g ∈ I(R3 ) so that gD0 ⊂ D1 or gD1 ⊂ D0 .
Proof.
vol{g ∈ I(R3 ) | gD1 ⊂ D0 or gD0 ⊂ D1 }= vol{g ∈ I(R3 ) | D0 ∩ gD1 ̸= ∅} − vol{g ∈ I(R3 ) | ∂D0 ∩ g∂D1 ̸= ∅}By Chern’s kinematic formula, we can estimate the ﬁrst term of the right hand side frombelow.
∫∫
N0dµ(g) ≥
χ(D0 ∩ gD1 )dµ(g){g∈I(R3 ) | D0 ∩gD1 ̸=∅}I(R3 )= 8π 2 (V0 χ(D1 ) + χ(D0 )V1 ) + 2π(F0 H̃1 + H̃0 F1 ).
By Fenchel’s theorem, the Poincaré formula and Chen’s kinematic formula, we can estimate the second term from above.
∫
2π
dµ(g)
{g∈I(R3 ) | ∂D0 ∩g∂D1 ̸=∅}(∫)∫≤κg ds dµ(g)I(R3 )∫≤) 21 (∫
1 ds

) 12

2

I(R3 )(∫≤=∂D0 ∩g∂D1(∫(∂D0 ∩g∂D1∂D0 ∩g∂D1

κ2g ds) 12 (∫
Length(∂D0 ∩ g∂D1 )dµ(g)I(R3 )

2π 3 F0 F1

) (
1
2

I(R3 )

dµ(g)
(∫

)

∂D0 ∩g∂D1

κ2g ds

) 12
dµ(g)

)
(2)
(2)
2π 3 (3H̃0 − K̃0 )F0 + 2π 3 F0 (3H̃1 − K̃1 ) 2 .
1

Therefore we can estimate vol{g ∈ I(R3 ) | gD1 ⊂ D0 or gD0 ⊂ D1 } from below. If
vol{g ∈ I(R3 ) | gD1 ⊂ D0 or gD0 ⊂ D1 } > 0, then there does exist g ∈ I(R3 ) such thatgD0 ⊂ D1 or gD1 ⊂ D0 . Thus we obtain the theorem.

6.2. Homologically volume minimizing submanifolds. In general, the Poincaré formulae, that is kinematic formulae for volume functional, can not be expressed as a constant times of the volumes of two submanifolds. However, in some cases we can estimate
the evaluation of the integral. Then, by that inequality, we can prove the homologically
volume minimizing property of certain submanifolds.
Let Gr (Rr+n ) denote the real Grassmannian manifold of all r-dimensional subspaces in
r+n
R . SO(r + n) acts on Gr (Rr+n ) transitively and it can be expressed as a homogeneousspace
Gr (Rr+n ) ∼= SO(r + n)/S(O(r) × O(n)).
Moreover Gr (Rr+n ) has an invariant Riemannian metric as a symmetric space. In her
paper [8], Lê Hông Vân obtained the following inequality.
10

Theorem 6.4 (Lê Hông Vân [8]). Let N be an rs-dimensional submanifold of Gr (Rr+n ).
Then∫vol(SO(r + n))#(N ∩ gGr (Rr+n−s ))dµSO(r+n) (g) ≤vol(N ).
vol(Gr (Rr+s ))SO(r+n)By this inequality, we can prove the following.
Theorem 6.5 (Lê Hông Vân [8]). Gr (Rr+s ) is a volume minimizing submanifold of
Gr (Rr+n ) in its Z2 -homology class.
Proof. Let N be a compact submanifold of Gr (Rr+n ) in the Z2 -homology class of Gr (Rr+s ).
Since #(Gr (Rr+s ) ∩ gGr (Rr+n−s )) = 1 for almost all g ∈ SO(r + n), we have#(N ∩ gGr (Rr+n−s )) ≡ 1 (mod 2)
for almost all g ∈ SO(r + n). Thus from Theorem 6.4 we have∫vol(SO(r + n)) ≤#(N ∩ gGr (Rr+n−s ))dµ(g)SO(r+n)=vol(SO(r + n))vol(N )
vol(Gr (Rr+s ))Hence we havevol(Gr (Rr+s )) ≤ vol(N ).

Furthermore, she proved homologically volume minimizing properties of sub-Grassmanniansof complex and quaternionic Grassmannian manifolds.
Theorem 6.6 (Lê Hông Vân [8]). Gr (Cr+s ) is a volume minimizing submanifold of
Gr (Cr+n ) in its Z2 -homology class.
Theorem 6.7 (Lê Hông Vân [8]). Gr (Hr+s ) is a volume minimizing submanifold of
Gr (Hr+n ) in its Z2 -homology class.
By the same method, Liu [9] proved that the Pontryagin cycle in SO(n) is homologically
volume minimizing in its homology class. And Tasaki [18] proved that the Helgasson
spheres in some symmetric spaces are homologically volume minimizing in their homology
classes.
6.3. Hamiltonian volume minimizing properties of Lagrangian submanifolds.
We apply the Poincaré formula to show Hamiltonian volume minimizing properties ofminimal Lagrangian submanifolds in some Kähler manifolds.
Let (M, ω) be a 2n-dimensional closed symplectic manifold with symplectic form ωand L be an n-dimensional closed submanifold of M . Then L is said to be Lagrangian if
ω|T L ≡ 0. Hamiltonian isotopies of (M, ω) are deﬁned as follows. If a smooth function
F : M × [0, 1] → R is given, then we can uniquely deﬁne the vector ﬁeld Xt on M foreach t ∈ [0, 1] such that
ω(Xt , ·) = dF (·, t).
Therefore, we have the ﬂow {ϕt }t∈[0,1] of diﬀeomorphisms on M deﬁned by the diﬀerential
equationdϕt (x) = Xt (ϕt (x))
dtwith initial condition ϕ0 = idM . The time 1-map ϕ1 of this ﬂow is called a Hamiltonian
diﬀeomorphism. The set of all Hamiltonian diﬀeomorphisms is denoted by Ham(M, ω).
11

Hereafter we restrict our attention to Kähler manifolds to introduce the volume functional. Let (M, ω, J) be a closed connected Kähler manifold. A Lagrangian submanifold
L in M is said to be Hamiltonian volume minimizing, if L has the least among all Hamiltonian deformations of L. This notion was ﬁrst introduced by Y. G. Oh [11], and hementioned only one non-trivial example:
Theorem 6.8 (Kleiner-Oh). The real projective space RP n canonically embedded in CP n
is a Hamiltonian volume minimizing Lagrangian submanifold.
In order to prove this theorem, we use the Poincaré formula for Lagrangian submanifolds
in CP n (Theorem 3.4), and the following Arnold-Givental inequality in the Lagrangianintersection theory.
Theorem 6.9 (Oh [12], [13] and [14]). Let (M, ω) be a compact symplectic manifold suchthat there exists an integrable almost complex structure J for which the triple (M, ω, J)
becomes a compact Hermitian symmetric space. Let L = Fix τ be the ﬁxed point set of ananti-holomorphic involutive isometry τ on M . Assume that the minimal Maslov numberof L is greater than or equal to 2. Then for any Hamiltonian diﬀeomorphism ϕ of M suchthat L and ϕ(L) intersect transversely, the inequality(6.1)♯(L ∩ ϕ(L)) ≥ SB(L, Z2 ) :=dim
∑LrankHi (L, Z2 )i=0holds.
Proof of Theorem 6.8. Let L be the real projective space RP n canonically embedded in
CP n . From Theorems 6.9 and 3.4, for any ϕ ∈ Ham(CP n , ωF S ) we have∫(n + 1)vol(U (n + 1))
vol(L)vol(ϕ(L)) ≥♯(L ∩ g ◦ ϕ(L))dµU (n+1) (g)(RP n )2
U (n+1)∫≥
SB(L, Z2 )dµU (n+1) (g)U (n+1)= vol(U (n + 1))SB(L, Z2 )= (n + 1)vol(U (n + 1)).
Hence we havevol(ϕ(L)) ≥ vol(L).

By the same method, we can prove the following theorem.
Theorem 6.10 (Iriyeh-Ono-S. [6]). A totally geodesic Lagrangian torus S 1 (1) × S 1 (1) in
S 2 (1) × S 2 (1) is Hamiltonian volume minimizing.
In this case, however Poincaré formula can not express as a constant times of thevolumes of two Lagrangian submanifolds, we can estimate the integral of intersection
numbers.
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Department of Mathematics and Information Sciences, Tokyo Metropolitan University, 1-1 Minami-osawa, Hachioji-shi, Tokyo, 192-0397 Japan
E-mail address: sakai-t@tmu.ac.jp13