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 $X$.
"Total curvature of noncompact piecewise Riemannian 2-polyhedra." Tsukuba J. Math. 29 (2) 471 - 493, December 2005. https://doi.org/10.21099/tkbjm/1496164966