Ehresmann connections, metrics and good metric derivatives. (English) Zbl 1144.53033

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, 263-308 (2007).
Authors’ abstract: In this survey we approach some aspects of tangent bundle geometry from a new viewpoint. After an outline of our main tools, i.e., the pull-back bundle formalism, we give an overview of Ehresmann connections and covariant derivatives in the pull-back bundle of a tangent bundle over itself. Then we define and characterize some special classes of generalized metrics. By a generalized metric we shall mean a pseudo-Riemannian metric tensor in our pull-back bundle. The main new results are contained in section 5. We shall say, informally, that a metric covariant derivative is ‘good’ if it is related in a natural way to an Ehresmann connection determined by the metric alone. We shall find a family of good metric derivatives for the so-called weakly normal Moór–Vanstone metrics and a distinguished good metric derivative for a certain class of Miron metrics.
For the entire collection see [Zbl 1130.53005].


53C05 Connections (general theory)
53C60 Global differential geometry of Finsler spaces and generalizations (areal metrics)