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
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