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Curvature measures and generalized Morse theory. (English) Zbl 0722.53064
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 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 application a theorem of M. Zähle is proved [Math. Nachr. 119, 327-339 (1984; Zbl 0553.60014)], extending the curvature measures to certain locally finite unions of sets of positive reach.

MSC:
53C65 Integral geometry
53A07 Higher-dimensional and -codimensional surfaces in Euclidean and related \(n\)-spaces
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