1. I N T R O D U C T I O N
Let G be a Lie group and K a closed subgroup of G. If M and N are compact subman-
ifolds of the homogeneous space GjK. Then a good deal of energy in integral geometry
has gone into computing integrals of the following type
(1-1) [ I{MngN)tla{g)
where J is an "integral invariant" of the submanifold M D gN. For example in the case
that G is the group of isometries of Euclidean space E n , M and N are submanifolds of
E n and I(M D gN) = Vol(Af H gN) then evaluation of (1-1) leads to formulas due to
Poincare, Blaschke and others (see the book [18] for references) or in the same case if we
let I(M H gN) be one of the integral invariants arising from the Weyl tube formula then
the evaluation of (1-1) gives the kinematic formula of Federer [8] and Chern [6]. In the
case G is the unitary group U(n -f 1) acting on complex projective space CP n and M and
N are complex analytic submanifolds of P n then letting I(M 0 gN) = Vol(M PI gN)
in (1-1) leads to results of Santalo [17] or letting I(M D gN) be the integral of a Chern
class leads to the recent kinematic formula of Shifrin [19]. In this paper we will assume
that GjK has an invariant Riemannian metric and evaluate (1-1) for arbitrary M and N
in the case that I(M H gN) = Vol(M D gN) (this generalizes the results of Brothers [2])
and for "arbitrary" integral invariants J in the case G is unimodular and acts transitively
on the sets of tangent spaces to each of M and N. That is we will give a definition of
integral invariant general enough to cover most cases that have come up to date and for
I(M DgN) one of these invariants we will evaluate (1-1) in terms of the integral invariants
of M and N. This leads to new formulas (at least modulo evaluating some constants) even
for submanifolds of Euclidean space E n .
Before giving a summary of our results we give a reasonably exact statement of our
results for submanifolds of Euclidean space. This should make what follows more concrete.
Recall that if
is a p dimensional submanifold of E
and x £ M then the second
fundamental form h^1 of M at x is a symmetric bilinear map from TMX x TMX to T^M*
(here TM is the tangent bundle of M and T±M is the normal bundle of M in E n ) .
If e\,..., e
is an orthonormal basis of E n such that e\,..., ep is a basis of TMX and
p + 1
, . . . , e
is a basis of T±MX then the components of h^f in this basis are the numbers
(^)?j ~
{h'x[(ei)ej)i e
«) 1 ~ hJ Pi V + 1 ^
where (,) is the usual inner product
on E n .
Received by the editor January 30, 1986.
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