Kinematic formulas in Riemannian spaces

Kinematic formulas in Riemannian spaces

Jiazu Zhou

Department of Mathematics, Southwest University,

Chongqing, 400715 and College of Mathematics and Statistics, Wuhan University, Wuhan, Hubei 430072, People’s Republic of China

e-mail : jzhou07828@yahoo.com

(2000 Mathematics Subject Classification : 52A22, 53C40, 53C65, 51C16.)

Abstract. LetGbe a Lie group andHits subgroup, and letM^p,N^qbe two submanifolds of dimensionsp,q, respectively, in the Riemannian homogeneous spaceG/H. LetI(M∩gN) be an integral invariant on the intersection submanifoldM∩gN and letdgbe the invariant density forG. The integralR

GI(M∩gN)dgexpressed by known invariants ofM andN is called the kinematic formula forI(M∩gN). By the method of Cartan moving frames, we will give a simplified proof of some known kinematic formulas for submanifolds in a Remannian space in this paper. The ideas can be useful for more general cases.

1 Introduction

Let G be a Lie group that acts on a smooth manifold X and H a compact subgroup ofG. Let dgbe an invariant density onG. We consider the the invariant measure that comes from the invariant densitydg. LetM^p,N^q be two submanifolds of dimensionsp,q, respectively, inG/H. We always assume thatM andN are in generic positions. This means that the dimension of the intersectionM∩gNis non- negative for almost allg∈G, that is, dim(M∩gN) =p+q−n≥0. LetI(M∩gN) be an integral invariant ofM∩gN. One of the basic problems in integral geometry is to find explicit formulas for integral of I(M ∩gN) over G in terms of known integral invariants of M and N. That is to find numerical constantscⁿ_pqi (depends on indices only), such that

(1)

Z

G

I(M∩gN)dg= X

0≤2i≤p+q−n

cⁿ_pqiI2i(M)In−2i(N),

whereIk(X) is an invariant ofX. This is called thekinematic formula forI(M∩gN) (see [1], [3], [5], [6], [7], [10], [11], [14], [15], [16], [17], [18] for references).

The kinematic formulas are not only interesting in their own cases but also have been applied to many mathematical branches and many geometric problems. For example, Grinberg, Ren and Zhou [8] obtained sufficient conditions for a domain to contain another in a plane of constant curvature by the fundamental kinematic

39

formula of Blaschk´e and Poincar´e. The sufficient condition for one domain to con- tain another in the Euclidean plane was first obtained by Hadwiger in 1941 (see [13], [16]). Zhou achieved an extension of Chen’s kinematic formula (see [3]) and applied it and obtained the Hadwiger’s containment condition for domains of higher dimensions (see [23], [26], [27], [28], [29]). Kinematic formulas have remarkable ap- plications in geometric convexity, geometric inequalities (see [21], [22], [30], [31]) and differential geometry (see [30]).

In the case of that Gis the group of isometries in the Euclidean spaceRⁿ, let M, N be two compact submanifolds in Rⁿ and I(M ∩gN) = Vol(M ∩gN) the volume ofM∩gNfor isometryg∈G. Then the following kinematic formula is due to Poincar´e, Blaschk´e and Santal´o (see [13], [16]):

(2)

Z

G

Vol(M ∩gN)dg= O_n· · ·O₁O_p+q−n

OpOq Vol(M)Vol(N),

whereOmis the volume of the unite sphere of dimensionminE^m+1, and its value is given by

Om= 2π^(m+1)/2 Γ((m+ 1)/2). Here Γ(·) is the Gamma function.

The formula is, respectively, due to Poincar´e and Blaschk´e forn= 2 andn= 3.

And it is given by Santal´o forn≥3.

(Poinca´e [13], [16]) Let Γ1, Γ2 be the rectifiable curves with lengths L1, L2, respectively. Then Z

G

#{Γ1∩gΓ2}dg= 4L1L2, where #{Γ1∩gΓ2} is the number of the intersection Γ1∩gΓ2.

LetMi (i= 1,2) be compact surface inR³ with mean curvatureHi and Gaus- sian curvatureKi, and let

H˜_i²= Z

Mi

H_i²dA, K˜i= Z

Mi

KidA,

where dAis the volume element. Let κbe the curvature of the intersection curve M1∩gM2andAithe area ofMianddsthe arc element. Then we have the following C-S Chen’s formula (see [3], [28]):

Z

G

µZ

M1∩gM2

κ²ds

¶

dg= 2πn³

3 ˜H₁²+ ˜K1

´ A2+³

3 ˜H₂²+ ˜K2

´ A1

o .

The fundamental formula of Blaschk´e (see [13], [16]) is

Proposition 1. Let Di (i= 1,2) be a domain in R³ with smooth boundary ∂Di, of which Vi, Ai, H˜i, χi are respectively the volume, the surface area, the integral

of mean curvature of boundary, and the Euler characteristic. Then if g ∈ G, the group of rigid motions of R³

Z

G

χ(D1∩gD2)dg= 8π²n

4π(χ1V2+χ2V1) + ˜H1A2+ ˜H2A1

o .

This formula was generalized to higher dimensional spaceRⁿ by Chern [6] and to the non-euclidean spaces by Santal´o [16].

LettingI(M∩gN) be one of the integral invariants arising from the Weyl tube formula will lead to the kinematic formula of Federer and Chern (see [7], [?]).

In the case of thatGis the unitary groupU(n+ 1) acting on complex projective spaceCPⁿ, andM andN are complex analytic submanifolds ofCPⁿ, then letting I(M∩gN) = Vol(M∩gN) leads to formulas of Santal´o. LettingI(M∩gN) be the integral of a Chern class leads to the kinematic formula of Shifrin [18].

For a homogeneous space G/H, letting I(M ∩gN) = Vol(M ∩gN) leads to formula of Howard (the generalized formula of Brothers [10]). Recently, Kang, Sakai and Suh have some concrete kinematic formulas for invariant polynomials of degree 2 on the second fundamental forms (see [11], [14], [15]).

The purpose of this paper is to give a simplified proof of the kinematic formula of Howard for invariant polynomials on the second fundamental forms. We will use the Cartan’s moving frames and directly derive the invariant polynomials on the second fundamental forms of the intersectionM∩gNas the homogeneous expression of the second fundamental forms ofM andN and theanglebetweenM andN.

We wish to thank Professor Young Jin Suh for his invitation to the 10th Interna- tional Workshop on Differential Geometry, held at Kyungpook National University, Taegu, November 10-11, 2005. We also like to thank all organizers of the work- shop and Professor Suh’s students for the great hospitalities during the successful workshop.

2 Preliminaries

LetX be ap-dimensional submanifold immersed in ann-dimensional Rieman- nian space N. We choose a local field of orthonormal framese1,· · ·, en in N such that, restricted to X, the vectore1,· · ·, ep are tangent to X. We make use of the following convention on the ranges of indices:

(3) 1≤A, B, C,· · · ≤n, p+ 1≤i, j, k,· · · ≤n, 1≤α, β, γ,· · · ≤p.

With respect to the frame field of N chosen above, let ω1,· · · , ωn be the field of dual frames. Then the structure equations ofN are given by

dx=X

A

ωAeA,

dω_A=−X

B

ω_AB∧ω_B, ω_AB+ω_BA= 0,

dωAB=−X

C

ωAC∧ωCB+ ΦAB, ΦAB= 1 2

X

C,D

KABCDωC∧ωD.

KABCD=KCDAB, KABCD=−KABDC =−KBACD, KABCD+KADBC+KACDB = 0.

If these are restricted to X, then ωi = 0. Since 0 =dωi =−P

αωiα∧ωα, by Cartan’s lemma we can write

ωiα=X

β

hⁱ_αβωβ, hⁱ_αβ=hⁱ_βα. From these formulas, we obtain

(4) dω_α=−X

β

ω_αβ∧ω_β, ω_αβ+ω_βα= 0,

(5) dωαβ=−X

γ

ωαγ∧ωγβ+ Ωαβ, Ωαβ= 1 2

X

γ,σ

Rαβγσωγ∧ωσ,

Rαβγσ=Rγσαβ, Rαβγσ=−Rαβσγ=−Rβαγσ, Rαβγσ+Rασβγ+Rαγσβ= 0, dωij =−X

k

ωik∧ωkj+ Ωij, Ωij =1 2

X

α,β

Rijαβωα∧ωβ,

Rijαβ =Rαβij, Rijαβ=−Rijβα=−Rjiαβ, Riαβγ+Riγαβ+Riβγα= 0.

The Riemannian connection ofX is defined by (ωαβ). The form (ωij) defines a connection in the normal bundle ofX. We call

II=X

i

IIiei=X

i

< d²x, ei> ei = X

i,α,β

hⁱ_αβωαωβei

the second fundamental formof the immersed submanifoldX. Sometimes we shall denote the second fundamental forms by

IIi=< d²x, ei >=X

α,β

hⁱ_αβωαωβ =< II, ei>

or simply its componentshⁱ_αβ. The length of the second fundamental formII ofX is defined by

|II|²=X

i

|IIi|²=X

i

X

α,β

¡hⁱ_αβ¢₂ .

Themean curvature vector−→

H is defined by

−

→H = 1 p

X

i

(trace(IIi))ei=1 p

X

i

ÃX

α

hⁱ_αα

! ei,

and its lengthH, that is,

H =1 p

(X

i

(trace(IIi))² )_1/2

=1 p



 X

i

ÃX

α

hⁱ_αα

!₂





1/2

is called themean curvatureofX.

Let V and W be vector subspaces of dimensional p and q, respectively. Let vp+1, . . . , vn be an orthonormal basis of N(V) and wq+1, . . . , wn an orthonormal basis ofN(W), that is,

N(V) = span{vp+1,· · · , vn};

(6)

N(W) = span{wq+1,· · · , wn},

the normal spaces to V, W, respectively. The angle between subspaces V andW is defined by

(7) ∆ =kvp+1∧ · · · ∧vn∧wq+1∧ · · · ∧wnk, where

(8) kx1∧ · · · ∧xkk²=|det(< xi, xs>)|.

IfV,W are both (n−1)-dimensional then ∆(V, W) =|sinθ|. It is obvious that 0≤∆≤1,

with

∆(V, W) = 0 if V ∩W 6={0}.

∆(V, W) = 1 if V ⊥W.

Also if gis a linear isometry of Eⁿ, then ∆(gV, gW) = ∆(V, W).

Let us list indices that we will use very often through the rest of this paper in the following table:

(9) 1≤A, B, C≤n; 1≤α, β≤p+q−n;

p+q−n+ 1≤i, j≤n; p+q−n+ 1≤a, b≤p;

1≤e, f ≤p; p+ 1≤λ, µ≤n;

p+q−n+ 1≤h, l≤q; 1≤u, v≤q; q+ 1≤ρ, σ≤n.

Let xeA be orthonormal frames, so that x ∈ M^p and e1,· · ·, ep are tangent to M^p at x. Similarly, let x⁰e⁰_A be frames, such that x⁰ ∈gN^q ande⁰₁,· · ·, e⁰_q are tangent to gN^q at x⁰. Suppose g be generic, so that M^p∩gN^q is of dimension p+q−n. We restrict the above families of frames by the condition

(10) x=x⁰, eα=e⁰_α.

Geometrically the latter means thatx∈M^p∩gN^q andeαare tangent toMp∩gN^q atx. The two submanifoldsM^p andN^q atxhave a scalar invariant, which is also called the “angle” betweenM^p andN^q, i.e.,

(11) ∆²=|det(eλ, e⁰_ρ)|.

In the case of that M^p and N^q are both hypersurfaces (p= q =n−1) it is the absolute value of the sine of the angle between their normal vector.

The second fundamental forms are all symmetric bilinear functions onTxM× TxM for all xin M. That is, the second fundamental form of M at x ∈M is a symmetric bilinear mapping

(12) h^M_x :Mx×Mx−→M_x^⊥,

where Mx is the tangent bundle of M and M_x^⊥ is the normal bundle of M at x.

If e1,· · · , en is orthonormal basis ofN such that e1,· · · , ep is a basis of Mx and ep+1,· · ·, en is a basis of M_x^⊥, then the components of h^M_x in this basis are the numbers¡

h^M_x ¢_i

αβ=< h^M_x (eα, eβ), ei>, 1≤α, β≤p,p+ 1≤i≤n.

3 Main theorems and the proofs

Call a polynomial P(xⁱ_αβ) in variablesxⁱ_αβ, 1≤α, β ≤p, p+ 1≤i ≤n and xⁱ_αβ=xⁱ_βα an invariant polynomial defined on the second fundamental forms ofp dimensional submanifolds, if it is invariant under the substitutions

xⁱ_αβ=X

s,t,j

aαsaβtx^j_stbij

for all orthogonal matrices (aαβ)p×pand (bij)(n−p)×(n−p). IfP is such a polynomial then

P(h^M_x ) =P³¡

h^M_x ¢_i

αβ

´

is defined independently of the choice of the orthonomal basice1,· · ·, en. For each such polynomial we define an integral invariant

IP(M) = Z

M

P(h^M_x )dA

By the invariance ofP with the substitution above it follows thatI_P is invariant for G, that is,IP(gM) =IP(M) for all isometriesgofRⁿ. We can now state Howard’s kinematic formula:

Theorem 1. LetM,N be respectively two compact submanifolds of dimensionsp,q in anndimensional spaceXⁿ of constant curvature such that for almost allg∈G, dim(M∩gN) =p+q−n≥0. LetP be a homogeneous invariant polynomial defined on the second fundamental forms of p+q−ndimensional submanifolds M ∩gN. If the degree of P is≤p+q−n, then there is a finite set of pairs (Qα,Rα) such that:

1. EachQα,Rα is respectively a homogeneous invariant polynomial defined on the second fundamental forms of M,N;

2. For eachα, degree(P) =degree(Qα) +degree(Rα);

3. Letdg be the invariant density on the group of isometries in Xⁿ, then Z

G

IP(M∩gN)dg=X

α

IQα(M)IRα(N).

To prove Howard’s theorem, let us prove the following theorem, which is useful and crucial for many cases. Actually Howard obtain his kinematic formula mainly depending on this theorem. We give a geometric and constructive proof.

Theorem 2. Let M^p,N^q be, respectively, a pair of submanifolds of dimensionsp, qin anndimensional Riemannian spaceNwithp+q−n≥0. Leth^λ_αβ,h⁰_αβ^ρ be the second fundamental forms of M^p,N^q, respectively. Let∆ be the angle betweenM^p andgN^q, forg∈G, the group of isometry ofN. LetII^g be the second fundamental form of the intersection submanifold M_g^p+q−n =M^p∩gN^q. Then we have

∆²II^g= P

λ,α,β

³

h^λ_αβ−P

σ aλσh⁰_αβ^σ´

ωαωβeλ

+P

ρ,α,β

³

h⁰_αβ^ρ −P

µ bρµh^µ_αβ´

ωαωβe⁰_ρ,

where aλσ andbρµ are angle elements betweenM^p andN^q.

Our goal is to express the second fundamental forms of the intersection ofp+ q−ndimensional manifoldM_g^p+q−n =M^p∩gN^q in terms of (homogeneous) those ofM^p andgN^q and the “angle” betweenM^pand gN^q.

We choose orthonormal frames{eA} and{e⁰_B} such that:

1. eα=e⁰_α;

2. e1,· · · , ep+q−n∈T(M^p∩gN^q);

3. e1,· · · , ep∈T(M^p);

4. e1,· · · , ep+q−n, e⁰_p+q−n+1,· · ·, e⁰_q ∈T(gN^q);

5. ep+1,· · ·, en∈N(M^p), the normal bundle of M^p; 6. e⁰_q+1,· · · , e⁰_n ∈N(gN^q), the normal bundle ofgN^q;

7. span{ep+1,· · ·, en, e⁰_q+1,· · · , e⁰_n}= span{ep+q−n+1,· · ·, ep, e⁰_p+q−n+1,· · ·, e⁰_q}

=N(M^p∩gN^q),the normal bundle of M^p∩gN^q. For the families of framesxeA andxe⁰_A, let

(13) ωA= (dx, eA), ω_A⁰ = (dx⁰, e⁰_A), (14) ωAB = (deA, eB), ω⁰_AB= (de⁰_A, e⁰_B), so that

(15) ωAB+ωBA= 0, ω⁰_AB+ω_BA⁰ = 0.

When restricted toM^p, N^q we have, respectively,

(16) ωλ= 0, ω_ρ⁰ = 0.

And restricted toM_g^p+q−n, we have

(17) ωαλ=X

β

h^λ_αβωβ, ω_αρ⁰ =X

β

h⁰_αβ^ρ ωβ,

where

(18) h^λ_αβ=h^λ_βα, h⁰_αβ^ρ =h⁰_βα^ρ ,

The second fundamental formsII^g ofM_g^p+q−n=M^p∩gN^q II^g=X

i

II_i^gei= X

i,α,β

hⁱ_αβωαωβei,

related to frames{e_A}, {e⁰_A}are, respectively II^g=X

a

IIaea+X

λ

IIλeλ; II^g=X

h

II_h⁰ e⁰_h+X

ρ

II_ρ⁰ e⁰_ρ,

where

(19) IIa = (d²x, ea) =P

α,βh^a_αβωαωβ; IIλ= (d²x, eλ) =P

α,βh^λ_αβωαωβ; II_h⁰ = (d²x, e⁰_h) =P

α,βh⁰_αβ^h ωαωβ; II_ρ⁰ = (d²x, e⁰_ρ) =P

α,βh⁰_αβ^ρ ωαωβ.

The submanifoldsM^pandgN^q have a scalar invariant, which is also the ”angle”

(module) betweenM^pandgN^q,

(20) ∆²=|det(ea, e⁰_ρ)|=|det(aρa)|=|det(eλ, e⁰_h)|=|det(bλh)|, aρa andbλh are the angle elements betweenM^pand N^q.

For a pair of hypersurfaces (p=q=n−1) it is clearly the absolute value of the cosine of the angle between their normal vectors.

We are now in the position to prove our theorems.

Proof. We wish to express (d²x, ea) as a linear combination of IIλ and II_ρ⁰. Therefore we set

(21) e⁰_ρ=X

a

aρaea+X

λ

aρλeλ

so that

(22) aρa= (e⁰_ρ, ea), aρλ= (e⁰_ρ, eλ).

Under our hypothesis ∆ =|det(aρa)| 6= 0. let (bbσ) be the inverse matrix of (aρa), so that

(23) X

σ

bbσaσa=δba, X

a

aρabaσ=δρσ. Then we have

(24) ea =X

ρ

baρe⁰_ρ+X

λ

baλeλ,

where

(25) baλ=−X

ρ

baρaρλ.

The condition (e⁰_ρ, e⁰_σ) =δ_ρσ is expressed by

(26) X

a

a_ρaa_σa+X

λ

a_ρλa_σλ=δ_ρσ. Therefore we have

(27) IIa= (d²x, ea) =X

α,β

h^a_αβωαωβ=X

ρ

baρII_ρ⁰ +X

λ

baλIIλ.

By the same way, we wish to express (d²x, e⁰_h) as a linear combination of IIλ

andII_ρ⁰. Therefore we set

(28) eλ=X

h

bλhe⁰_h+X

σ

bλσe⁰_σ,

so that

(29) bλh= (eλ, e⁰_h), bλσ = (eλ, e⁰_σ).

Under our hypothesis ∆ =|det(bλh)| 6= 0. let (alµ) be the inverse matrix of (bλh), so that

(30) X

λ

ahλbλl=δhl, X

h

bλhahµ=δλµ. Then we have

(31) e⁰_h=X

λ

ahλeλ+X

σ

ahσe⁰_σ,

where

(32) ahσ=−X

λ

ahλbλσ.

The condition (eλ, eµ) =δλµ is expressed by

(33) X

l

bλlbµl+X

σ

bλσbµσ=δλµ. Therefore we have

(34) II_h⁰ = (d²x, e⁰_h) =X

α,β

h⁰_αβ^hωαωβ =X

σ

ahσII_σ⁰ +X

λ

ahλIIλ.

To express the second fundamental forms of M_g^p+q−n as a linear combination ofIIλ andII_ρ⁰, we set

(35) II^g = X

λ,α,β

X_αβ^λ ωαωβeλ+ X

ρ,α,β

Y_αβ^ρ ωαωβe⁰_ρ,

where X_αβ^λ andY_αβ^0ρ are to be determined.

Therefore, we have

II^g= X

λ,α,β

X_αβ^λ ωαωβeλ+ X

ρ,α,β

Y_αβ^ρ ωαωβ

ÃX

a

aρaea+X

λ

aρλeλ

! (36)

= X

a,α,β

ÃX

ρ

aρaY_αβ^ρ

!

ωαωβea+ X

λ,α,β

Ã

X_αβ^λ +X

ρ

aρλY_αβ^ρ

!

ωαωβeλ.

Hence we have

½ h^a_αβ = P

ρaρaY_αβ^ρ ; h^λ_αβ = X_αβ^λ +P

ρa_ρλY_αβ^ρ . (37)

Similarly, we have II^g= X

λ,α,β

X_αβ^λ ωαωβ

ÃX

h

bλhe⁰_h+X

σ

bλσeσ

!

+ X

σ,α,β

Y_αβ^σ ωαωβe⁰_σ (38)

= X

h,α,β

ÃX

λ

bλhX_αβ^λ

!

ωαωβe⁰_h+ X

σ,α,β

Ã

Y_αβ^σ +X

λ

bλσX_αβ^λ

!

ωαωβe⁰_σ,

and (

h^0h_αβ = P

λbλhX_αβ^λ ; h^0σ_αβ = Y_αβ^σ +P

λbλσX_αβ^λ . (39)

Combining above equations together gives ( h^λ_αβ = X_αβ^λ +P

ρaλρY_αβ^ρ ; h⁰_αβ^ρ = Y_αβ^ρ +P

λbρλX_αβ^λ , (40)

or

à h^λ_αβ h⁰_αβ^ρ

!

=

µ (Iλλ) (aλρ) (bρλ) (Iρρ)

¶ µ X_αβ^λ Y_αβ^ρ

¶ . (41)

Finally, we have µ X_αβ^λ

Y_αβ^ρ

¶

=

µ (Iλλ) (aλρ) (bρλ) (Iρρ)

¶₋₁Ã h^λ_αβ h⁰_αβ^ρ

! (42)

= 1

∆²

µ (Iλλ) (−aλρ) (−bρλ) (Iρρ)

¶ Ã h^λ_αβ h⁰_αβ^ρ

! ,

where ∆²=det (Iλλ) (aλρ) (bρλ) (Iρρ) . That is

∆²X_αβ^λ =h^λ_αβ−P

σ aλσh⁰_αβ^σ,

∆²Y_αβ^ρ =h⁰_αβ^ρ −P

µ bρµh^µ_αβ. (43)

Inserting (43) into (35) we complete the proof of Theorem 2.

Let M, N be two hypersurfaces in the Euclidean space Rⁿ. We choose the frames {e_A} and {e⁰_A} such that e₁ = e₁,· · · , e_n−2 = e⁰_n−2 are tangent to Σ_g = M∩gNanden,e⁰_nare, respectively, the normal vector ofM,N. The angle between M andN is ∆ =|sinφ|andaλσ =bρµ= cosφ. Then we have the following special and very important case.

Theorem 3. Let M, N be two hypersurfaces of class C² in the Euclidean space Rⁿ and lethⁿ_ij,h⁰_ijⁿ be the normal curvatures ofM,N, respectively. Then we have

(44) sin²φ IIΣg =



X

i,j

hⁿ_ij−cosφX

i,j

h⁰_ijⁿ



en+



X

i,j

h⁰_ijⁿ−cosφX

i,j

hⁿ_ij



e⁰_n,

wherecosφ= (en, e⁰_n).

By taking the normal we have the following generalized Euler formula (see [4], [19], [24])

Theorem 4. Let M,N be two hypersurfaces of classC² inRⁿ and let hⁿ_ij,h⁰_ijⁿ be the normal curvatures ofM,N, respectively. Then we have

(45) sin²φ¯

¯IIΣg

¯¯²=



X

i,j

hⁿ_ij





2

+



X

i,j

h⁰_ijⁿ





2

−2 cosφ



X

i,j

hⁿ_ij







X

i,j

h⁰_ijⁿ



,

wherecosφ= (e_n, e⁰_n).

If M, N ⊂R³ are two smooth surfaces, we choose the frames {e1, e2, e3} and {e⁰₁, e⁰₂, e⁰₃} such that e1 = e⁰₁, the tangent of the curve Γg = M ∩gN, for rigid motiong ∈ G, ande3, e⁰₃ are, respectively the normal of M, N. Let κ^M_n and κ^N_n be, respectively the normal curvatures of M and N. Then we immediately obtain (also see [24])

Theorem 5. LetM,N be two smooth surfaces inR³and letκ^M_n ,κ^N_n be the normal curvatures ofM,N, respectively. Then we have

(46) sin²φ IIΓg =¡

κ^M_n −κ^N_n cosφ¢ e3+¡

κ^N_n −κ^M_n cosφ¢ e⁰₃, where cosφ= (e3, e⁰₃).

Note¯

¯IIΓg

¯¯=κ, the curvature of Γg. Then by taking the norm we immediately obtain the following known classical Euler formula.

Theorem 6. Let M andN be two surfaces inR³ with the normal curvaturesκ^M_n andκ^N_n. Letκbe the curvature of the intersection curveM∩gN andφbe the angle between M andgN. Then we have the following Euler formula ([4], [19], [24])

κ²sin²φ=¡ κ^M_n ¢2

+¡ κ^N_n¢2

−2 cosφ¡ κ^M_n ¢ ¡

κ^N_n¢ .

By Theorem 2, any invariant homogeneous polynomial defined on the second fundamental forms ofM^p∩gN^q can be expressed as the linear combination of the invariant homogeneous polynomials defined on the second fundamental forms ofM^p andN^q.

Integrating over the density formula (see [6]) of Chern Θgdg=±∆^k+1Θ∆ΘMΘN

(where Θ is the density) will leads to Howard’s kinematic formula.

The case of p =q = n−1 is very important and the constants in Howard’s kinematic formula could be determined and the invariants on the right hand side could be those jth order integral of mean curvature ofM andN.

For an n dimensional submanifoldM inRⁿ, if we pick any pair of independent tangent vectorsuandv inTp(M), the tangent space atp∈M, then for every unit vector w =λu+µv, there is a unique geodesic in M starting at p with tangent vector w. The set of all such geodesics, as w describes the unit circle in the plan spanned by u and v, sweep a surface whose Gauss curvature at p is the section curvatureKΠ=K[u, v] of the plane Π spanned byuandv.

Lete1,· · ·, embe an orthonormal basis of the tangent spaceTp(M) of M atp.

Then the quantity

(47) S= 2 X

1≤i<j≤m

K[ei, ej]

is independent of the choice of basis, and is called the scalar curvature ofM atp.

For a hypersurface Σ in Rⁿ, we may choose e₁,· · ·, e_n−1 to be the principal curvature directions at p. Then the scalar curvature S of Σ can be expressed in terms of the principal curvaturesκ1,· · · , κn−1 by

(48) S= 2 X

1≤i<j≤n−1

κiκj.

One consider the Gauss map

(49) G:p7→N(p)

whose differential

(50) dGp:x⁰(t)7→N⁰(t)

wherex(t) =p, satisfies Rodriques’ equations

(51) dGp(ei) =−κiei, i= 1,2,· · · , n−1.

Therefore we have the mean curvature

(52) H = 1

n−1(κ1+· · ·+κn−1) =− 1

n−1trace(dGp) and the Gauss-Kronecker curvature

(53) K=κ1· · ·κn−1= (−1)ⁿ⁻¹det(dGp).

Thejth-order mean curvature is the jth-order elementary symmetric functions of the principal curvatures. We denote byHj thejth-order mean curvature, normal- ized such that

(54)

n−1Y

j=1

(1 +tκj) =

n−1X

j=0

µn−1 j

¶ Hjt^j.

That is

(55) Hj=

µ n−1 j

¶₋₁

{κi1,· · ·, κij}; j= 1,· · ·, n−1

where {κi1,· · · , κij} is the j-th elementary symmetric functions of the principal curvatures. Thus,H1=H, the mean curvature, andHn−1 is the Gauss-Kronecker curvature. For n = 3, that is all about it, but in higher dimensions, there are intermediateHj. Among them,H2plays an important role in differential geometry.

Therefore we have

(56) S= (n−1)(n−2)H2.

Thej-th-order integral of mean curvatureM_j(Σ) is defined by

(57) Mj(Σ) =

Z

Σ

HjdA, j= 1,· · ·, n−1,

where dA is the volume element of Σ. We let M0(Σ) = F, the area of Σ for completeness.

If Σ is a convex hypersurface bounding a convex bodyKinRⁿ, we have the re- lations between integrals of mean curvatures of Σ(≡∂K) andj-th-order Minkowski quermassintegralsWj ofK (see [13], [16], [17])

(58) Mj(Σ) =nWj+1(K); j = 0,1,· · ·, n−1.

Note that Minkowski quermassintegralsWj are well defined for any convex figure, whereasMj(∂K) makes sense only if∂K is of classC². In this case, we have the total scalar curvature

(59) S˜=

Z

Σ

SdA=n(n−1)(n−2)W3(K).

And we have the following Alexandrov-Fenchel inequalities W_i²≥Wi−1Wi+1; W1= A

n; W0=V, (i= 1,· · ·, n−1),

whereAandV are, respectively, the area and the volume ofK. The equality holds whenK is a standard ball.

The Howard’s kinematic formula for the case of hypersurfaces can be obtained directly via mean curvature integrals of boundaries∂Dk of domains Dk (k=i, j).

We are not going to details here.

References

[1] J. E. Brothers, Integral geometry in homogeneous space, Trans. Amer. Math.

Soc. 124 (1966), 480-517.

[2] B-Y Chen, Geometry of Submanifolds, Marcel Dekker. Inc., New York (1973).

[3] C-S. Chen, On the kinematic formula of square of mean curvature, Indiana Univ. Math. J., 22 (1972-73), 1163-1169.

[4] F. Chen, X. Zhao & J. Zhou, An anlogue of the Euler formula, J. of Math., 25 (2005) 1-4.

[5] S. S. Chern, On the kinematic formula in the euclidean space of n dimensions, Amer. J. Math., 74 (1952), 227-236.

[6] S. S. Chern, On the kinematic formula in integral geometry, J. of Mathematics and Mechanics, 16 (1966), 101-118.

[7] H. Federer, Curvature measure, Trans. Amer. Math. Soc., 69 (1959), 418-491.

[8] E. Grinberg, D. Ren & J. Zhou, The Symmetric Isoperimetric Deficit and the Containment Problem in a Plan of Constant Curvature, preprint.

[9] E. Grinberg, G. Zhang, J. Zhou & S. Li, Integral Geometry and Convexity, World Scientific, Singapore (2005).

[10] R. Howard, The kinematic formula in riemannian geometry, Memoirs of Amer.

Math. Soc., 509 (1993).

[11] H.J. Kang, T. Sakai & Y.J. Suh, Kinematic formulas for integral invariants of degree two in real space froms, Indiana Univ. Math. J., 54(2005), 1499-1519.

[12] D. Klain and G-C. Rota, Introduction to Geometric Probability, Cambridge University Press, (1997)

[13] D. Ren, Topics in Integral Geometry, Series in Pure Mathematics, World Sci- entific (Singapore), 19 (1994).

[14] T. Sakai, Integral geometry and Hamiltonian volume minimizing propety, Pro- ceedings of the 9th International Workshop on Diff. Geom., 9 (2005), 69-84.

[15] T. Sakai, Poincar´e formula in irreducible Hermitian symmetric spaces, Tokyo J. Math. 26 (2003), no. 2, 541-547.

[16] L. A. Santal´o, Integral Geometry and Geometric Probability. Addison-Wesley, Reading, Mass. (1976).

[17] R. Schneider, Convex Bodies: The Brunn-Minkowski Theory, New York: Cam- bridge University Press, (1993)

[18] T. Shifrin, The kinematic formula in complex integral geometry, Trans. Amer.

Math. Soc., 264 (1981) No. 2, 255-293.

[19] M. Spivak, A Comprehensive Introduction to Differential Geometry, Publish or Perish Inc., (1979).

[20] H. Weyl, On the volume of tubes, Amer. J. Math., 61 (1939), 461-472.

[21] G. Zhang, A Sufficient Condition for one Convex Body Containing Another, Chin. Ann. of Math., 9B-4 (1988), 447-451.

[22] G. Zhang & J. Zhou, Containment measures in integral geometry, Integral geometry and Convexity, World Scientific, Singapore (2005).

[23] J. Zhou, Kinematic formulas for mean curvature powers of hypersurfaces and analogous of Hadwiger’s theorem in R²ⁿ, Trans. Amer. Math. Soc., 345 (1994),243-262.

[24] J. Zhou, Kinematic formula for square mean curvature of hypersurfaces, Bull.

of the Institute of Math. (Academia Sinica), 22 (1994), 31-47.

[25] J. Zhou, On the Willmore deficit of convex surfaces, Lectures in Applied Math- ematics of Amer. Math. Soc., 30 (1994), 279-287.

[26] J. Zhou, When can one domain enclose another inR³? J. Austral. Math. Soc.

(series A), 59 (1995), 266-272.

[27] J. Zhou, The sufficient condition for a convex domain to contain another in R⁴, Proc. Amer. Math. Soc., 121 (1994), 907-913.

[28] J. Zhou, A kinematic formula and analogous of Hadwiger’s theorem in space, Contemporary Mathematics, 140 (1992), 159-167.

[29] J. Zhou, Sufficient conditions for one domain to contain another in a space of constant curvature, Proc. Amer. Math. Soc., 126 (1998), 2797-2803.

[30] J. Zhou, On the Willmore functional for hypersurfaces, submitted.

[31] J. Zhou, On Bonnesen-type isoperimetric inequalities, Proceedings of the 10th International Workshop on Diff. Geom. (2005).