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VOL. 48 | 2007 Finsler geometry in the tangent bundle

Abstract

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.

Information

Published: 1 January 2007
First available in Project Euclid: 16 December 2018

zbMATH: 1144.53094
MathSciNet: MR2389255

Digital Object Identifier: 10.2969/aspm/04810163

Subjects:
Primary: 53C60

Rights: Copyright © 2007 Mathematical Society of Japan

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