On Finsler spaces with Randers’ metric and special forms of important tensors

Abstract

I n 1941 G. Randers introduced first a special Finsler metric cis =(g i i (x)dxidxj) 1 / 2 b i (x)dxi , i n a viewpoint o f general relativity [1 2 ]*. Since then many physicists have developed th e general relativity based o n this m etric. (See References of [5]). F ro m t h e standpoint o f Finsle r geom etry itself R anders' metric is very interesting, because its form is sim ple and properties o f th e Finsler space equipped w ith this m etric m u st be described by th e ones of the R iem annian space equipped w ith t h e metric L(x, d x)=(g u (x)dx idx j)'/ 2 together w ith t h e 1-form 13(x, dx)=b i (x) d x l. F o r example th e curvature tensors R h i i k , P h u , a n d S", i , o f t h e Finsler space m u s t b e w ritte n in term s o f R iem annian tensors, that is , t h e curvature te n so r, b i a n d its covariant derivatives w ith respect t o t h e Riem annian connection. B ut w e have few papers concerned w ith t h e Finsler space i n viewpoint of F insler geom etry [4 ], [5 ], [1 0 ], [1 3 ]. T h is s itu a tio n se e m s to c o m e fro m t h e f a c t th a t w e m u s t h i t a t o n c e against insuperable difficulty o f e x h a u stin g c a lc u la tio n s to o b ta in t h e c o n c re te fo rm o f Cartan's T h e p u rp o se o f t h e p r e s e n t p a p e r is to w r ite th e to rsio n a n d curvature tensors o f t h e R anders space (t h e Finsler space equipped * Num bers in brackets refer to the references at the end of the paper.