Intrinsic curvatures in analytic-geometric categories
type: Item
instance of: scholarly article
title: Intrinsic curvatures in analytic-geometric categories
zbMATH Open document ID: 1053.53053
author: Andreas Bernig
author: Ludwig Broecker
publication date: 2004-01-06T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/116088
review text: 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)].
reviewed by: Enrique Macias-Virgós
Mathematics Subject Classification ID: 53C65
Mathematics Subject Classification ID: 14P10
zbMATH DE Number: 2021765
zbMATH Keywords: curvatures
zbMATH Keywords: subanalytic spaces
zbMATH Keywords: kinematic formula
zbMATH Keywords: densities
MaRDI profile type: Publication
OpenAlex ID: W2314337117
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DOI: 10.5802/AIF.1995
published in: Annales de l’institut Fourier
description: scientific article; zbMATH DE number 2021765