0091 on S - 3 like Four - Dimensional Finsler Spaces

نویسنده

  • P. N. PANDEY
چکیده

In 1977, M. Matsumoto and R. Miron [9] constructed an orthonormal frame for an n-dimensional Finsler space, called ‘Miron frame’. The present authors [1, 2, 3, 10, 11] discussed four-dimensional Finsler spaces equipped with such frame. M. Matsumoto [7, 8] proved that in a three-dimensional Berwald space, all the main scalars are h-covariant constants and the h-connection vector vanishes. He also proved that in a three-dimensional Finsler space satisfying T-condition, all the main scalars are functions of position only and the v-connection vector vanishes [6, 7]. The purpose of the present paper is to generalize these results for an S-3 like four-dimensional Finsler space. 1. Preliminaries Let M be a four-dimensional smooth manifold and F 4 = (M, L) be a fourdimensional Finsler space equipped with a metric function L(x, y) on M. The normalized supporting element, the metric tensor, the angular metric tensor and Cartan tensor are defined by li = ∂̇iL, gij = 1 2 ∂̇i∂̇jL , hij = L∂̇i∂̇jL and Cijk = 1 2 ∂̇kgij respectively. The torsion vector C i is defined by C = C jkg . Throughout this paper, we use the symbols ∂̇i and ∂i for ∂/∂y i and ∂/∂x respectively. The Cartan connection in the Finsler space is given as CΓ = (F i jk, G i j , C i jk). The hand v-covariant derivatives of a covariant vector Xi(x, y) with respect to the Cartan connection are given by (1.1) Xi|j = ∂jXi − (∂̇hXi)G h j − F r ijXr, and (1.2) Xi|j = ∂̇jXi − C r ijXr. 2000 Mathematics Subject Classification. 53B40.

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تاریخ انتشار 2010