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" .
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