Introduction

It is our goal in this article to present a current and uniform treatment of flag and Ricci curvatures in Finsler geometry, highlighting recent developments. (The flag curvature is a natural extension of the Riemannian sectional curvature to Finsler manifolds.) Of particular interest are the Einstein metrics, constant Ricci curvature metrics and, as a special case, constant flag curvature metrics. Our understanding of Einstein spaces is inchoate. Much insight may be gained by considering the examples that have recently proliferated in the literature. This motivates us to discuss many of these metrics.

Happily, the theory is developing as well. The Einstein and constant flag curvature metrics of spaces of Randers type, a fecund class of Finsler spaces, are now properly understood. Enlightenment comes from being able to identify the class as solutions to Zermelo's problem of navigation, a perspective that allows a very apt characterisation of the Einstein spaces. When specialised to flag curvature, the navigation description yields a complete classification of the constant flag curvature Randers metrics.

We hope to bring out the rich variety of behaviour displayed by these metrics. For example, Finsler metrics of constant flag curvature exhibit qualities not found in their constant sectional curvature Riemannian counterparts.

• • Beltrami's theorem guarantees that a Riemannian metric is projectively flat if and only if it has constant sectional curvature. On the other hand, there are many Finsler metrics of constant flag curvature which are not projectively flat. See Section 3.2.3 and [Shen 2004].