On Generalised Recurrent Finsler Spaces

Abstract

In one of his papers Vranceanu [9] has defined a non-symmetric connection in an n- dimensional space , we extend this concept to the theory of n- dimensional Finsler space with non-symmetric connection based on a non-symmetric fundamental tensor . Let us write (1.1) = + , Where and are respectively the symmetric and skew-symmetric parts of Following Cartan [1], let a vertical stroke ( | ) followed by an index denote covariant derivative with respect to x, here we define the covariant derivative of any contravariant vector field as follows: (1.2) = - ( + where, a positive sign below an index and followed by a vertical stroke indicates that the covariant derivative has been formed with respect to the connection as for as that index is concerned. The covariant derivative defined in (1.2) will be called - covariant differentiation of with respect to . Differentiating (1.2) - covariantly with respect to and taking the skew-symmetric part of the result so obtained with respect to indices j and k, we obtain the following commutation formula (1.3) - = - ( + + where (1.4) - + - + - . The entities defined by (1.4) is called “ Curvature Tensor” of the Finsler space equipped with nonsymmetric connection. From here onwards the Finsler space equipped with non-symmetric connection will be denoted by . We shall extensively use the following identities, notations and contractions: