Tsukuba Journal of Mathematics

Total curvature of noncompact piecewise Riemannian 2-polyhedra

Jin-ichi Itoh and Fumiko Ohtsuka

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In this paper, we treat piecewise Riemannian 2-polyhedra which are combinatorial 2-polyhedra such that each 2-simplex is isometric to a triangle bounded by three smooth curves on some Riemannian 2-manifold. We will introduce the total curvature $C(X)$ of a piecewise Riemannian 2-polyhedron $X$ not only in the compact case but also in the noncompact case, and obtain some generalizations of the Gauss-Bonnet theorem and the Cohn-Vossen theorem. Furthermore, we will show that the difference between $C(X)$ and some value concerning to the topology of $X$ coincides with some expanding growth rate of $X$.

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Tsukuba J. Math., Volume 29, Number 2 (2005), 471-493.

First available in Project Euclid: 30 May 2017

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Itoh, Jin-ichi; Ohtsuka, Fumiko. Total curvature of noncompact piecewise Riemannian 2-polyhedra. Tsukuba J. Math. 29 (2005), no. 2, 471--493. doi:10.21099/tkbjm/1496164966. https://projecteuclid.org/euclid.tkbjm/1496164966

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