Curvature measures and generalized Morse theory
type: Item
instance of: scholarly article
title: Curvature measures and generalized Morse theory
zbMATH Open document ID: 0722.53064
published in: Journal of Differential Geometry
publication date: 1989-01-01T00:00:00Z
review text: In studying the differential geometry of a smooth hypersurface \(M^ n\) in Euclidean space \(E^{n+1}\) it is fruitful to view the integral of the Gauss-Kronecker curvature as an integral over the unit sphere \(S^ n\), i.e. as the area of the Gauss map \(\nu\). A further step identifies the value of the integrand on the sphere, at some point \(v\in S^ n\), with the sum of some topological indices associated to the height function \(h_ v(x):=x\cdot v,\) \(x\in M\) and to the points of \(\nu^{-1}(v)\). In fact these latter points are exactly the critical points of this height function and the topological index at each point is \((-1)^{\lambda}\), where \(\lambda\) is the Morse index of \(h_ v\) there. This relates the total absolute curvature of compact \(M^ n\) to the sum of its Betti numbers [cf. the work of S. S. Chern-R. K. Lashof, N. H. Kuiper, Th. F. Banchoff). The indices above may exist even when \(M^ n\) is not smooth. On the other side there are Federer's curvature measures for sets A of positive reach in Euclidean spaces. In the present paper the author extends the elementary notions of Morse theory to \(C^{1,1}\)-hypersurfaces and to sets of positive reach; the strategy is to compare the behavior of functions on A to the behavior of associated functions on tubular neighborhoods of A. Then the main theorem expresses the Gauss curvature measure \(\Phi_ 0(A,)\) as follows: for any Borel set K \[ (n+1)\alpha (n+1)\Phi_ 0(A,K)=\int_{S^ n}\sum_{p\in K\cap A,-v\in nor(A,p)}(-1)^{\lambda}dH^ nv, \] where \(\alpha (n+1)\) is the volume of the unit ball in \(E^{n+1}\), \(H^ n\) is the n-dimensional Hausdorff measure, and \(\lambda\) is the Morse index of the height function \(h_ v\), (cf. 4.6.1. Thm. in \textit{M. Zähle} [Geom. Dedicata 23, 155-171 (1987; Zbl 0627.53053)]. Also similar expressions for the curvature measures \(\Phi_ i\) are given (Cauchy formula: reproductive property of the curvature measures through intersections with i-planes). Moreover as an appli…
Mathematics Subject Classification ID: 53C65
Mathematics Subject Classification ID: 53A07
zbMATH DE Number: 4190643
zbMATH Keywords: Morse theory
zbMATH Keywords: sets of positive reach
zbMATH Keywords: Gauss curvature
zbMATH Keywords: height function
zbMATH Keywords: curvature measures
author: Joseph H. G. Fu
reviewed by: Eberhard Teufel
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.4310/jdg/1214443826
OpenAlex ID: W1511103951
Wikidata QID: Q115182470
DOI: 10.4310/JDG/1214443826
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description: scientific article; zbMATH DE number 4190643