Linear and metrical connections of a Riemannian space, whoseindicatrices are ellipsoids, are established in the tangent bundle. In-dicatrices of Finsler spaces are smooth, starshaped and convex hyper-surfaces. They do not transform, in general, into each other by lineartransformations, and thus they do not admit linear metrical connec-tions in the tangent bundle. This necessitates the introduction of line-elements yielding the dependence of the geometric objects not only ofpoints x but also of the direction y. Therefore, the apparatus (con-nections, covariant derivatives, curvatures, etc.) of Finsler geometrybecomes 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 impor-tant special Finsler spaces, which allow linear metrical connections inthe tangent bundle. In this paper we want to present results which use the tangent bun-dle TM only, and do not need TT/1/ or VT M or line-elements. Theseinvestigations often admit direct geometrical considerations. Longerproofs are only sketched or omitted.
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