A variational proof of the Gauss-Bonnet formula |
| |
Authors: | Dr. Moritz Armsen |
| |
Affiliation: | (1) Abteilung Mathematik, Universität Dortmund, 4600 Dortmund 50, West Germany |
| |
Abstract: | The classical Gauss-Bonnet formula has the form I(gij)=2, where I(gij) represents a sum of three terms each of which depends on the metric tensor gij. It is shown that the first variation I of I(gij) with respect to the metric gij vanishes and that for the Euclidean metric ij we have I(ij)=2. From this the formula I(gij)=2 follows. In the process, explicit expressions are obtained for the first variation of each of the three terms which comprise I(gij). Furthermore, a general expression for the first variation of a multiple integral whose integrand is a scalar density depending on the metric tensor gij and its derivatives up to the second order is obtained with the aid of results of Rund [1]. |
| |
Keywords: | |
本文献已被 SpringerLink 等数据库收录! |
|