INTRODUCTION 3
on the full group of motions G, but only on the invariant theory of the isotropy subgroup
K. This this observation leads to a "transfer principle" allows us to "move" kinematic
formulas proven for a homogeneous space G/K to any other homogeneous space with an
isotropy subgroup equivalent to K. For example, the Chern-Federer kinematic formula for
submanifolds of E n is
(1-4) / wiM'ngN^Slaig) = Y^c{n,P,q,l,k)li2k{M^Kl_k){N")
where the ^'s are the integral invariants from the Weyl tube formula (defined in section 10
below), G is the group of isometries of
Rn
and c{n,p, g, /, k) is a constant only depending on
the indicated parameters. The transfer principle tells us this formula holds in all simply
connected spaces of constant sectional curvature (the sphere and the hyperbolic space
form) with the same values for the constants c(n,p, g,Z, k). (Here the integrand in the
definition of fA2k{M) must be expressed—and this is an important point—as a polynomial
in the components of the second fundamental form of M and not as a polynomial in the
components of the curvature tensor of M.)
We now summarize our results. In section 2 we prove our "basic integral formula"
for submanifolds M and N of a Lie group G on which all our latter integral formulas
will be based. The idea behind its proof is extremely simple: Apply the Federer coarea
formula to the function f : M x N —* G, given by f(^rj) = £T7 - 1 , and interpret the
result geometrically. Although the details are quite different the proofs are very much
in the style of the papers of Federer [8] and Brothers [2] to which the present paper is
greatly indebted. One big difference between the proofs here and those in [2] and [8] is
that we work in the smooth category and thus avoid the measure theoretic problems which
Federer and Brothers have to deal with. In section 3 the integral (1-1) is evaluated for
any compact submanifolds M and N of a Riemannian homogeneous space G/K in the
case I(M D gN) = Vol(M fl gN) and examples are given of how the transfer principle in
this context can be used to compute the various constants occurring in the formulas in an
efficient and elegant manner. The proofs here precede by applying the integral formula of
section 2 to the submanifolds 7r - 1 M and 7r-1iV of G (where 7r : G —* G/K is the natural
projection) and then "pushing" the result of this back down to G/K. In an appendix
to this section a general Crofton type formula is proven. Apart from its own interest
this Crofton formula lets us to identify the invariant measures used in Chern's paper [4]
with Riemannian invariants. This allows examples to be given of homogeneous spaces (in
particular CF 2 ) where these measures are different from the Riemannian volume of the
submanifold and where these measures are not unique, so that the choice of the measure
to be used is determined by the type of integral geometric formula to be proven.
In section 4 we give a general definition of an integral invariant of a compact subman-
ifold M of a Riemannian homogeneous space G/K. Once this has been done the general
kinematic formula and the transfer principle are stated and in an appendix a general ana-
logue of the "linear" kinematic formulas in section 8 of [6] and section 3 of [19] is given.
The next two sections contain the lemmas needed to prove these results. In particular,
section 5 gives the needed results on the geometry of intersections of submanifolds and
section 6 gives the required algebraic facts and definitions.
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