Riemann-Finsler geometry is a subject that concerns manifolds with Finsler metrics, including Riemannian metrics. It has applications in many fields of the natural sciences. Curvature is the central concept in Riemann–Finsler geometry. This invaluable textbook presents detailed discussions on important curvatures such as the Cartan torsion, the S-curvature, the Landsberg curvature and the Riemann curvature. It also deals with Finsler metrics with special curvature or geodesic properties, such as projectively flat Finsler metrics, Berwald metrics, Finsler metrics of scalar flag curvature or isotropic S-curvature, etc. Instructive examples are given in abundance, for further description of some important geometric concepts. The text includes the most recent results, although many of the problems discussed are classical.

Contents:

• Finsler Metrics
• Structure Equations
• Geodesics
• Parallel Translations
• S-Curvature
• Riemann Curvature
• Finsler Metrics of Scalar Flag Curvature
• Projectively Flat Finsler Metrics

“This is a concise book written for graduate students and young geometers who are interested in Riemann–Finsler geometry.”

“This interesting text is a concise well organized textbook concerning Riemann–Finsler geometry. It is characterized by its clarity and simplicity, on one hand, and its rigor, on the other hand ... this is an excellent book that should take its place in any mathematical library.”