## Tsukuba Journal of Mathematics

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

### Total curvature of noncompact piecewise Riemannian 2-polyhedra

Jin-ichi Itoh and Fumiko Ohtsuka

#### Abstract

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 $X$.

#### Article information

**Source**

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

**Dates**

First available in Project Euclid: 30 May 2017

**Permanent link to this document**

https://projecteuclid.org/euclid.tkbjm/1496164966

**Digital Object Identifier**

doi:10.21099/tkbjm/1496164966

**Mathematical Reviews number (MathSciNet)**

MR2177022

**Zentralblatt MATH identifier**

1104.53039

#### Citation

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