The Total Curvature (Curvatura Integra) of a Closed Surface with Riemannian Metric and Poincaré’s Theorem on the Singularities of Fields of Line Elements

Abstract

A line element on a surface S is determined by a non-zero tangent vector to the surface. The same line element is determined by all non-zero multiples of the vector. Hence there is no distinguished direction on a line element. Strictly speaking, a line element is a one dimensional linear subspace of the tangent vector space.