Finsler geometry in the tangent bundle. (English) Zbl 1144.53094

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, 163-194 (2007).
The present work goes deep to the basics of Finsler spaces, by investigating fundamental geometric objects such as distance, angle and connections. In the first part, it studies the relation between distance spaces and Finsler spaces, providing an outlook on the properties of distance functions induced by Finsler structures, and of Finsler structures determined by distance functions. In the second part, conditions are provided by Minkowskian angle which imply the reduction of a Finsler space to a Riemannian one; as well, it is proved that a diffeomorphism between two Finsler spaces is an isometry provided that it is angle and area-preserving. In the last part affine deformations of Finsler spaces are examined and the local Minkowski structures, and it is pointed out which other Finsler spaces – besides Euclidean, Riemannian, Minkowskian and locally-Minkowskian, still allow linear metrical connections on their tangent space, and what are the special features of their geometry.
For the entire collection see [Zbl 1130.53005].


53C60 Global differential geometry of Finsler spaces and generalizations (areal metrics)
53B05 Linear and affine connections
53B40 Local differential geometry of Finsler spaces and generalizations (areal metrics)