4 THE KINEMATIC FORMULA IN RIEMANNIAN HOMOGENEOUS SPACES

In section 7 a restatement of the kinematic formula is given in terms of the algebraic

definitions of section 6. This new form of the kinematic formula makes the results of section

4 quite transparent and is better adapted to concrete calculations. This theorem is then

proven. As with the results of section 3 the proof proceeds by replacing the submanifolds

M and N of G/K by 7T"1 M and TT~1N (n : G — » G/K natural projection), using the basic

integral formula of section 2, and then pushing the result back down to G/K.

The next section gives a proof that for spaces of constant sectional curvature the integrals

involved in equation (1-3) converge when degree('P) p + q — n + l. The main tool in the

proof is a formula from Chern's paper [6].

The last two sections of the paper are devoted to giving a new proof of the Chern-

Federer kinematic formula (1-4) which works in all simply connected spaces of constant

sectional curvature. This could be done by using the transfer principle to "move" the

result from R n , where it is known, to the other space forms. However, this does not lead

to any new insights. The idea in our proof of (1-4) is to give an algebraic characterization

of the polynomials appearing as the integrands of the /x's, which is of interest in its own

right, and which exhibits both the Weyl tube formula and (1-4) as consequences of the

invariant theory of the orthogonal group.

In an appendix we give a short proof of the coarea formula for smooth maps which

avoids the measure theoretic complications arising in the case of Lipschitz maps.

It is worth remarking at this point that the methods used here seem to be best adapted to

proving integral geometric formulas involving purely Riemannian invariants. For example

it is possible to give a proof of the main result of Shifrin's paper [19] in the style of the

proof given here of the Chern-Federer kinematic formula. This can be done (at least for

complex hypersurfaces) by giving a characterization of the integral invariants arising in

the formula for the volume of a tube about a complex analytic submanifold of C P n similar

to the one given in section 9 below for the /i's (see [23], [10] or [12] for the tube formula

in CP n ). The resulting formula, is in terms of integrals over the submanifolds of invariant

polynomials in the components of the second fundamental forms. But then one of the

prettiest facts about these invariants becomes almost invisible, they are also integrals of

Chern forms which represent cohomology classes on the submanifolds. The proof in [19]

not only makes this clear, it uses this fact strongly in the proof. On the other hand, there

are Riemannian integral invariants Iv of complex analytic submanifolds of C P n which are

not covered by the theorems in [19] (he only considers invariant polynomials in the Chern

forms and the Kaehler form) for which (1-1) can be evaluated by the methods given here.

Our notation and terminology is standard. By "smooth" we mean of class C°°. If M

is a smooth manifold then TM is its tangent bundle and TMX its tangent space at x. If

/ : M — N is a. smooth map between manifolds then f*x : TMX — TNf^ is the derivative

of / at x 6 M. If M and N are Riemannian manifolds then / : M — » N is a Riemannian

submersion iff for all x £ M the derivative f*x : TMX — TNf(x) is surjective and /*x

restricted to the orthogonal complement of kernel/*^ is a linear isometry. In the case

dim(M) — dim(iV) then a Riemannian submersion is a local isometry. We regard discrete

subsets S of a manifold as submanifolds of dimension zero in which case the volume of S is

defined to be the number of points in S. Lastly, if / : M — » T V is an immersed submanifold

of M , then we will repress the immersion / and just say that "TV is a submanifold of M" .