GaussBonnetTheorem 2007-01-22 17:41:55 2007-05-12 21:44:48 Theorem % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here (Carl Friedrich Gauss and Pierre Ossian Bonnet) Given a two-dimensional compact Riemannian manifold $M$ with boundary, Gaussian curvature of points $G$ and geodesic curvature of points $g_x$ on the boundary $\partial M$, it is the case that \[ \int_M G \, dA + \int_{\partial M}g_x ds = 2\pi\chi(M), \] where $\chi(M)$ is the Euler characteristic of the manifold, $dA$ denotes the measure with respect to area, and $ds$ denotes the measure with respect to arclength on the boundary. This theorem expresses a topological invariant in terms of geometrical information.