Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)


SIGMA 3 (2007), 118, 11 pages      arXiv:0709.4376      http://dx.doi.org/10.3842/SIGMA.2007.118
Contribution to the Proceedings of the 2007 Midwest Geometry Conference in honor of Thomas P. Branson

On Gauss-Bonnet Curvatures

Mohammed Larbi Labbi
Mathematics Department, College of Science, University of Bahrain, 32038 Bahrain

Received August 27, 2007, in final form November 15, 2007; Published online December 11, 2007

Abstract
The (2k)-th Gauss-Bonnet curvature is a generalization to higher dimensions of the (2k)-dimensional Gauss-Bonnet integrand, it coincides with the usual scalar curvature for k =1. The Gauss-Bonnet curvatures are used in theoretical physics to describe gravity in higher dimensional space times where they are known as the Lagrangian of Lovelock gravity, Gauss-Bonnet Gravity and Lanczos gravity. In this paper we present various aspects of these curvature invariants and review their variational properties. In particular, we discuss natural generalizations of the Yamabe problem, Einstein metrics and minimal submanifolds.

Key words: Gauss-Bonnet curvatures; Gauss-Bonnet gravity; lovelock gravity; generalized Einstein metrics; generalized minimal submanifolds; generalized Yamabe problem.

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