Intrinsic curvatures in analytic-geometric categories
The notions of \textit{Quermassintegrale} of a convex set (Minkowski) and of \textit{invariant volumes} of a submanifold of Euclidean space (Weyl) are generalized in the paper under review to sets having strong singularities and are embedded in a (possibly not flat) Riemannian manifold. The so-called relative and intrinsic \textit{Lipschitz-Killing curvatures} are defined by the authors in the setting of analytic-geometric categories introduced by \textit{L. van den Dries} and \textit{C. Miller} [Duke Math. J. 84, No. 2, 497--540 (1996; Zbl 0889.03025)], thus providing a very general formalism. A key ingredient of the definitions is the notion of \textit{normal cycle} introduced by \textit{J. H. G. Fu} [Indiana Univ. Math. J. 38, No. 3, 745--771 (1989; Zbl 0668.49010)]. Among the applications, the authors prove that relative curvatures are related to the volume growth in manifolds of constant sectional curvature. In particular they prove a generalization of the Gauss-Bonnet theorem and a kinematic formula which completes a result from Santaló [\textit{L. A. Santaló}, Introduction to integral geometry. (Paris: Hermann (1953; Zbl 0052.39403)], see also \textit{H. G. Fu} [Indiana Univ. Math. J. 39, No. 4, 1115--1154 (1990; Zbl 0703.53059)].
scientific article; zbMATH DE number 2021765
scientific article