On Gauss-Bonnet Curvatures

TL;DR

This paper reviews higher-dimensional Gauss-Bonnet curvatures, their variational properties, and explores their applications in generalizations of geometric problems like the Yamabe problem and Einstein metrics.

Contribution

It provides a comprehensive review of Gauss-Bonnet curvatures, highlighting their variational aspects and applications in higher-dimensional geometry and physics.

Findings

Analysis of variational properties of Gauss-Bonnet curvatures

Extensions of the Yamabe problem to higher dimensions

Connections to Einstein metrics and minimal submanifolds

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.