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Variation of curvatures of subanalytic spaces and Schläfli-type formulas. (English) Zbl 1037.32012
The aim of this paper is to prove a variational formula for Lipschitz-Killing curvatures which applies to a large class of singular spaces, namely that of subanalytic sets.
The method to obtain the main result uses the normal cycle construction. This normal cycle in some sense encodes the singularities of \(X\). This construction is made by using stratified Morse theory. Then the Lipschitz-Killing invariants can be obtained by integration of some differential form over the normal cycle.
As a nice consequence of this result Schläfi’s formula, Hilbert’s variational formula for the total scalar curvature of a manifold and the Chern-Gauss-Bonnet theorem can be put into a very general setting where these classical results can be obtained as special cases of the single variational formula proved in this paper.

MSC:
32C05 Real-analytic manifolds, real-analytic spaces
32B20 Semi-analytic sets, subanalytic sets, and generalizations
53C65 Integral geometry
53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
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