Advanced Studies in Pure Mathematics
- Adv. Stud. Pure Math.
- Finsler Geometry, Sapporo 2005 — In Memory of Makoto Matsumoto, S. V. Sabau and H. Shimada, eds. (Tokyo: Mathematical Society of Japan, 2007), 163 - 194
Finsler geometry in the tangent bundle
Linear and metrical connections of a Riemannian space, whose indicatrices are ellipsoids, are established in the tangent bundle. Indicatrices of Finsler spaces are smooth, starshaped and convex hypersurfaces. They do not transform, in general, into each other by linear transformations, and thus they do not admit linear metrical connections in the tangent bundle. This necessitates the introduction of line-elements yielding the dependence of the geometric objects not only of points $x$ but also of the direction $y$. Therefore, the apparatus (connections, covariant derivatives, curvatures, etc.) of Finsler geometry becomes inevitably a little more complicated.
Nevertheless there are a number of problems which need no line-elements. Such are those, which concern the metric only (arc length, area, angle, geodesics, etc.) and also the investigation of those important special Finsler spaces, which allow linear metrical connections in the tangent bundle.
In this paper we want to present results which use the tangent bundle $TM$ only, and do not need $TTM$ or $VTM$ or line-elements. These investigations often admit direct geometrical considerations. Longer proofs are only sketched or omitted.
Received: 11 February 2006
Revised: 17 July 2006
First available in Project Euclid: 16 December 2018
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Tamássy, Lajos. Finsler geometry in the tangent bundle. Finsler Geometry, Sapporo 2005 — In Memory of Makoto Matsumoto, 163--194, Mathematical Society of Japan, Tokyo, Japan, 2007. doi:10.2969/aspm/04810163. https://projecteuclid.org/euclid.aspm/1545001106