In this paper, we study Zermelo navigation on Riemannian manifolds and use that to solve a long standing problem in Finsler geometry, namely the complete classification of strongly convex Randers metrics of constant flag curvature.

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Zermelo navigation on Riemannian manifolds

Theory of Finsler spaces with (α, β)-metric

Finsler metrics on an open subset in R n with straight geodesics are said to be projective. It is known that the flag curvature of any projective Finsler metric is a scalar function of tangent vectors (the flag curvature must be a constant if it is Riemannian). In this paper, we discuss the classification problem on projective Finsler metrics of constant flag curvature. We express them by a Taylor expansion or an algebraic formula. Many examples constructed in this paper can be used as models in Finsler geometry.

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Projectively flat Finsler metrics of constant flag curvature

Introduction 198 1. Flag and Ricci Curvatures 199 1.1. Finsler metrics 199 1.2. Flag curvature 207 1.3. Ricci curvature 214 2. Randers Metrics in Their Defining Form 218 2.1. Basics 218 2.2. Characterising Einstein–Randers metrics 220 2.3. Characterising constant flag curvature Randers metrics 228 3. Randers Metrics Through Zermelo Navigation 230 3.1. Zermelo navigation 230 3.2. Navigation description of curvature conditions 236 3.3. Issues resolved by the navigation description 241 4. Einstein Metrics of Nonconstant Flag Curvature 244 4.1. Examples with Riemannian–Einstein navigation data 244 4.2. Examples with Kähler–Einstein navigation data 247 4.3. Rigidity and a Ricci-flat example 251 5. Open Problems 254 5.1. Randers and beyond 254 5.2. Chern’s question 255 5.3. Geometric flows 256 Acknowledgments 256 References 256

Ricci and Flag Curvatures in Finsler Geometry

This article uses the Berwald connection exclusively, together with its two curvatures, to cut an efficient path across the landscape of Finsler geometry. Its goal is to initiate differential geometers into two key research areas in the field: the search for unblemished “unicorns” and the study of Ricci flow. The exposition is almost self-contained.

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Makoto Matsumoto

Papers

On Finsler spaces with the curvature tensors Phijk and Shijk satisfying special conditions

The corrected fundamental theorem on the randers spaces of constant curvature

On Einstein's gravitational field equation in a tangent Riemannian space of a Finsler space

Finsler spaces with the hv-curvature tensor Phijk of a special form

The length and the relative length of tangent vectors of finsler spaces