A Note on the Gauss-Bonnet Theorem for Finsler Spaces

Abstract

Exactly fifty years ago, one of us gave a proof of the Gauss-Bonnet formula for Riemannian manifolds by the method of transgression ([Chl], [Ch2]), and introduced a 'total curvature' H whose properties have yet to be fully exploited. Other proofs have since been given, including one as the simplest case of the Atiyah-Singer index theorem. The Finsler side of the story is what concerns us in the following pages, and it begins with a work of Lichnerowicz's [L] in 1948. In that paper, using the Cartan connection, Lichnerowicz established a Gauss-Bonnet theorem for all Finsler surfaces [modulo the issue of Vol(x) discussed below] and also for all Cartan-Berwald spaces of even dimension greater than two. His proof was modelled after the intrinsic method just mentioned. There are several interesting issues raised by Lichnerowicz's paper. One concerns the volume Vol(x) of the unit Finsler sphere IxM in each tangent space TxM. Various attempts to understand why he assigned these volumes the constant Euclidean values (as in Riemannian geometry) have led to some developments which play a key role in our treatment here. There is also the issue which revolves around the choice of a connection. It appears that his restriction to Cartan-Berwald spaces was dictated by the structure of the curvature tensor of the Cartan connection. We will show that the use of a connection introduced by one of us [Ch3] in 1948 effects the extension of Lichnerowicz's result to a much larger class of Finsler manifolds. Finally, there is the question of where the Gauss-Bonnet integrand should live. In Riemannian geometry, it is a top degree form on the underlying manifold M. It was Lichnerowicz who proposed that for the Finsler case, little is lost by allowing this integrand to live on the projective sphere bundle SM, as