A variational proof of the Gauss-Bonnet formula

Abstract

The classical Gauss-Bonnet formula has the form I(g_ij)=2π, where I(g_ij) represents a sum of three terms each of which depends on the metric tensor g_ij. It is shown that the first variation δI of I(g_ij) with respect to the metric g_ij vanishes and that for the Euclidean metric δ_ij we have I(δ_ij)=2π. From this the formula I(g_ij)=2π follows. In the process, explicit expressions are obtained for the first variation of each of the three terms which comprise I(g_ij). Furthermore, a general expression for the first variation of a multiple integral whose integrand is a scalar density depending on the metric tensor g_ij and its derivatives up to the second order is obtained with the aid of results of Rund [1].