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Curvature measures of subanalytic sets. (English) Zbl 0818.53091
The Lagrangian cycle for general subanalytic sets in real analytic manifolds have been constructed by M. Kashiwara and P. Schapira in their book [Sheaves on manifolds (Springer-Verlag 1990; Zbl 0709.18001)]. The author of this paper is inspired by the description, given by Vintgen and Zähle (see the references of the paper under review), of the classical curvature integrals (Blaschke, Federer, Banchoff) as integral currents in the sense of Federer and Fleming. The special feature of the subanalytic category allows him to construct conormal cycles by approximation. In fact, the notion of conormal cycles is associated to the zero-sets of certain non-negative functions (among them subanalytic ones). What is interesting is that some version of the classical kinematic formula for subanalytic sets in \(\mathbb{E}^ n\) is proved using some “localization” of the Gauss-Bonnet theorem, which is important for the general uniqueness result obtained by means of the sophisticated technique developed in the author’s previous papers. Of course, the Gauss-Bonnet theorem plays the crucial role. The mentioned uniqueness theorem yields by a straightforward computation the Kashiwara- Schapira result. Finally, the curvature properties of subanalytic sets are studied. The notion of curvature measure is defined on the base of the geodesic curvature forms of Chern and it is proved that the top curvature measure is invariant under subanalytic isometries.
Reviewer: S.Dimiev (Sofia)

MSC:
53C65 Integral geometry
53C20 Global Riemannian geometry, including pinching
32B20 Semi-analytic sets, subanalytic sets, and generalizations
58C25 Differentiable maps on manifolds
58C35 Integration on manifolds; measures on manifolds
58A25 Currents in global analysis
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