Non-existence of certain types of NP-Finsler spaces
A projective Finsler space \(F_ n\) satisfying the relation \(\nabla_{\ell}N^ h_{ijk}=\lambda_{\ell}N^ h_{ijk}\) (resp. \(\nabla_{\ell}N^ h_{ijk}=0)\) for a non-vanishing covariant vector field \(\lambda_{\ell}(x)\), where \(\nabla\) is the operator of covariant differentiation with respect to the normal projective connection \(\Pi^ i_{jk}(x,\dot x)\) of \(F_ n\) with \(N^ h_{ijk}\) is the corresponding curvature tensor with respect to \(\Pi^ i_{jk}\), is called a recurrent (resp. symmetric) normal projective Finsler space, denoted by \(RNP\)-F\({}_ n\) (resp. \(SNP\)-F\({}_ n)\) [see \textit{R. B. Misra} and \textit{F. M. Meher}, Tensor, New Ser. 22, 117-120 (1971; Zbl 0206.509); the reviewer, Mat. Vesn. 10(25), 105-110 (1973; Zbl 0268.53009)]. In this paper, the author calls such a Finsler space for which Berwald's connection coefficients \(G^ i_{jk}\) also satisfy \(G^ i_{jkr}=0\) trivial normal projective Finsler space, and shows that a non-trivial RNP- (resp. SNP-) Finsler space \(F_ n\) \((n>2)\) does not exist.
scientific article; zbMATH DE number 4017793
scientific article