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On the curvature of piecewise flat spaces. (English) Zbl 0559.53028

The main result is: Let \(R^ j(U)\) be the integral of the j-th Lipschitz-Killing curvature of a Riemannian manifold \(M^ n\) over a neighborhood U bounded in \(M^ n\). Let \(M^ n_{\eta}\) be a piecewise linear approximation of \(M^ n\) of fatness bounded away from zero and \(R^ j_{\eta}\) the combinatorially defined j-th Lipschitz-Killing curvature. Then under certain regularity conditions for the approximations, \(| R^ j(U)-R^ j_{\eta}(U)| \to 0\) as \(\eta\) \(\to 0\). The corresponding pointwise result is false. The proof involves constructions in linear algebra and combinatorial topology that are interesting in themselves. The paper also contains a great number of illuminating comments, and an extension of the result to manifolds with boundary.
Reviewer: H.Guggenheimer

MSC:

53C20 Global Riemannian geometry, including pinching
57Q55 Approximations in PL-topology
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