1. Introduction

Roughly speaking, Finsler metrics on a manifold are regular, but not necessarily reversible, distance functions. In 1854, B. Riemann attempted to study a special class of Finsler metrics—Riemannian metrics—and introduced what is now called the Riemann curvature. This infinitesimal quantity faithfully reveals the local geometry of a Riemannian manifold and becomes the central concept of Riemannian geometry. It is a natural problem to understand general regular distance functions by introducing suitable infinitesimal quantities. For more than half a century, there had been no essential progress until P. Finsler studied the variational problem in a Finsler manifold. However, it was L. Berwald who first successfully extended the notion of Riemann curvature to Finsler metrics by introducing what is now called the Berwald connection. He also introduced some non-Riemannian quantities via his connection [Berwald 1926; 1928]. Since then, Finsler geometry has been developed gradually.

The Riemann curvature is defined using the induced spray, which is independent of any well-known connection in Finsler geometry. It measures the shape of the space. The Cartan torsion and the distortion are two primary geometric quantities describing the geometric properties of the Minkowski norm in each tangent space. Differentiating them along geodesics gives rise to the Landsberg curvature and the S-curvature. These quantities describe the rates of change of the “color pattern” on the space.

In this article, I am going to discuss the geometric meaning of the Landsberg curvature, the S-curvature, the Riemann curvature, and their relationship. I will give detailed proofs for several important local and global results.