2

THE KINEMATIC FORMULA IN RIEMANNIAN HOMOGENEOUS SPACES

Call a polynomial V(X%) in variables JVg 1 ij p, p 4- 1 a n and X* = X £

which is invariant under the substitutions

(1-2) Xfj -- ] T *isajtXPtba(3

s,t,(3

for all p by p orthogonal matrices [a{j] and all (n—p) by (n—p) orthogonal matrices [&a^] an

invariant polynomial defined on the second fundamental forms of p dimensional

submanifolds. If V is such a polynomial then

is defined independently of the choice of the orthonormal basis e i , . . . , e

n

. For each such

polynomial define an integral invariant V* on compact p dimensional submanifolds of R

n

by

F(M)= f v(h*J)nM(x)

JM

where fi^ is the volume density on M. Using the invariance of V under the substitution (1-

2) it follows that Iv has the basic invariance property Iv(gM) = I^(M) for all isometries

g of W1. This set of invariants contains a large number of the integral invariants which

occur in geometry.

We now state the kinematic formula:

T h e o r e m . Let p, q he integers with 1 p,q n and p + q n. Let V be an invariant

polynomial defined on the second fundamental forms ofp + q — n dimensional submanifolds

and assume that V is homogeneous of degree p + q — n + 1. Then there is a finite set of

pairs (Q«,7^

a

) such that:

(1) Each Qa is a homogeneous invariant polynomial on the second fundamental forms

of p dimensional submanifolds,

(2) Each 'R.ft is a homogeneous invariant polynomial on the second fundamental forms

of q dimensional submanifolds,

(3) For each a degree Q

a

4- degree 7Za — degree V,

(4) For all compact p dimensional submanifolds M and q dimensional submanifolds N

of¥in (each possibly with boundary)

(i-3) / iv(Mn9N)nG(g) = £ / C a ( M ) J * a (*)

where G is the group of isometries of E n and 0 ^ its invariant measure.

Once the group theoretic ideas involved in proving this have been isolated it becomes no

harder to prove (1-3) for submanifolds M and N of an arbitrary Riemannian homogeneous

space G/K provided only that G is unimodular and G is transitive on the sets of tangent

spaces to each of M and N. One advantage to working in this generality is that it becomes

clear that the form of kinematic formulas in a homogeneous space G/K does not depend