Some properties for Weyl’s projective Curvature Tensor of Generalized \(W^h\)-Birecurrent in Finsler Space

Abstract

In this paper, we defined a Finsler space \(F_n\) for which Weyl’s projective curvature tensor \(W_{jkh}^i\) satisfies the generalized-birecurrence condition with respect to Cartan’s connection parameters \(\Gamma_{kh}^{*i}\) , given by the condition \(W_{jkh׀l׀m}^i = α_{lm} W_{jkh}^i + β_{lm} (δ_h^i g_{jk} - δ_k^i g_{jh})\), where \(׀l׀m\) is h-covariant derivative of second order ( Cartan’s second kind covariant differential operator ) with respect to \(x^l\) and \(x^m\), successively, \(α_{lm}\) and \(β_{lm}\) are non-null covariant vectors field and such space is called as a generalized \(W^h\)-birecurrent space and denoted briefly by \(G W^h-BRF_n\) . We have obtained the h-covariant derivative of the second order for Wely’s projective torsion tensor \(W_{kh}^i \), Wely’s projective deviation tensor \(W_h^i\) and Weyl’s projective curvature tensor \(W_{jkh}^i\) and some tensors are birecurrent in our space. We have obtained the necessary and sufficient condition for Cartan’s third curvature tensor \(R_{jkh}^i\) , the associate curvature tensor \(R_{jpkh}\) to be generalized birecurrent, the necessary and sufficient condition of h-covariant derivative of second order for the h(v)-torsion tensor \(H_{kh}^i\) , the associate torsion tensor \(H_{kp.h}\) and the deviation tensor \(H_h^i\) has been obtained in our space.