Riemann-Finsler surfaces. (English) Zbl 1144.53093

Sabau, Sorin V. (ed.) et al., Finsler geometry, Sapporo 2005. In memory of Makoto Matsumoto. Proceedings of the 40th Finsler symposium on Finsler geometry, Sapporo, Japan, September 6–10, 2005. Tokyo: Mathematical Society of Japan (ISBN 978-4-931469-42-6/hbk). Advanced Studies in Pure Mathematics 48, 125-162 (2007).
The paper provides the Gauss-Bonnet theorem for Finsler Landsberg surfaces with smooth boundary, a natural extension of both the classical Gauss-Bonnet theorem for Riemannian surfaces with smooth boundary, and for the Gauss-Bonnet theorem for boundaryless Finsler surfaces [D. Bao and S. S. Chern, Ann. Math. (2), 143, No. 2, 233–252 (1996; Zbl 0849.53046)]. Comprehensive self-contained preliminaries on the geometry of Riemann-Finsler surfaces are provided as well, encompassing: basic properties of Minkowski planes, the Riemannian length of the indicatrix of a Minkowski norm, the Chern connection on Finslerian surfaces, Landsberg and Berwald surfaces, and the study of the geodesic curvature tensor and of the signed curvature of curves on Finsler surfaces.
For the entire collection see [Zbl 1130.53005].


53C60 Global differential geometry of Finsler spaces and generalizations (areal metrics)
53B40 Local differential geometry of Finsler spaces and generalizations (areal metrics)
53C40 Global submanifolds


Zbl 0849.53046