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Curvature of singular spaces via the normal cycle. (English) Zbl 0796.53070
Greene, Robert (ed.) et al., Differential geometry. Part 2: Geometry in mathematical physics and related topics. Proceedings of a summer research institute, held at the University of California, Los Angeles, CA, USA, July 8-28, 1990. Providence, RI: American Mathematical Society. Proc. Symp. Pure Math. 54, Part 2, 211-221 (1993).
The idea of associating with certain subsets \(X\) of Euclidean space a current \(N(X)\), called the normal cycle of \(X\), which yields in a natural way the generalized curvatures of \(X\), was first introduced by Wintgen and Zähle. The author announces the results and some of the methods of a program to study the normal cycles of sets \(X\) defined by analytic equations and maps, which do not generally admit decompositions of the type studied by Wintgen and Zähle. The key point is to characterize directly the normal cycle of a singular subspace, without using the method of Wintgen and Zähle. The normal cycle of an “analytic” subset may then be constructed in a straightforward way. The author then shows that certain invariants constructed from the normal cycle have meanings analogous to those of classical curvatures.
For the entire collection see [Zbl 0773.00023].

53C65 Integral geometry
32C30 Integration on analytic sets and spaces, currents