Euler Chracteristics of Surfaces with Negative Curvature

Abstract

In this paper,we show that the influence of Gaussian curvature penetrates to the topological conformation of M, when it is a surface in R³- to properties independent of the geometry of M. We introduce the Gauss-Bonnet Theorem on compact oriented surfaces and its applications. Finally, we show that a compact surface has negative Euler characteristic if and only if it admits a metric of constant negative curvature.