Dynamics Seminar

Date: Wednesday, February 8, 2012.
Room and time: Groningen, Zernike Campus, Room 5161.0293 (Bernoulliborg), 15:00.

Mathijs Wintraecken

Department of Mathematics, University of Groningen

On the Gauss-Bonnet theorem

Abstract

The Gauss-Bonnet theorem relates local differential geometrical quantities, that is quantities which depend only on the Riemannian metric and its derivatives, by integration to a global topological invariant, namely the Euler characteristic (\(\chi\)). It is not directly obvious that the Euler characteristic is the only quantity which can be established in this manner. We shall prove for a surface that if f is a function that can be expressed locally in terms of the metric and all its derivatives such that \[ \int_M f dA = t \] where \(t\) is a topological invariant, then \(t = c \chi\) for some constant \(c\). We shall generalize this statement to three dimensional manifolds and prove that under the same conditions \(t = 0\) and touch on problems occurring in dimensions exceeding three.