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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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