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On geodesic mappings of \(m\)-symmetric and generally semi-symmetric spaces. (English. Russian original) Zbl 0816.53009
Russ. Math. 36, No. 8, 38-42 (1992); translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1992, No. 8(363), 42-46 (1992).
This article is devoted to strengthening and generalizing a number of results of the same author investigating the problem of non-trivial geodesic mappings (abbreviate NGM). In [Math. Publ. (Opava). 1, 151-156 (1993; Zbl 0805.53061)], Ž. Radulovič and the author established that non-flat \(V_ n\) whose Riemann tensor satisfies (1) \(R_{hijk,\ell_ 1,\dots,\ell_ m} = R_{hi\alpha \beta} S^{\alpha \beta}\) (\(S\) is a tensor), do not admit NGM under the condition (2) \(a_{ij,k} = \lambda_{(i}g_{j)k}\), \(\lambda_{ij} = \mu g_{ij}\), \(\mu\) is a constant, where \(g_{ij}\) is the metric tensor, \(a_{ij}\) is a symmetric tensor, \(\lambda_ i\) is a non-zero vector and \((i,j)\) is symmetrization.
The main theorem of this paper is a generalization of a result of the article quoted above, replacing (1) by the existence of a tensor \(T_{h_ 1,\dots, h_ m}\); \(n \geq 2m-1\) not representable as a tensor sum of products of invariants and the metric tensor which satisfies: \[ \sum^ m_{s = 1} T_{h_ 1 \dots h_{s-1}} \alpha h_{s + 1} \dots h_ m(R^ \alpha_{h_ sjk} - B(\delta^ \alpha_ j g_{h_ sk} - \delta^ \alpha g_{hsj})) \] where \(\delta^ h_ j\) are Kronecker symbols and \(B\) is invariant. The spaces in which this condition is satisfied have been called \(T\)-generalized semi-symmetrical \(\{TP_ n(B)\}\) and \(T\)-semi-symmetrical \(\{TP_ n(0)\}\). It is demonstrated that an \(m\)-symmetrical \(V_ n\) \((n \geq 2m + 3)\) of non- constant curvature or a Riemannian space \(V_ n\) \((n \geq 2m)\) which satisfies: \((R_{_ ij,k} - R_{ik,j})_{e_ 1,\dots,e_ m} = 0\), \(R_{ij,k} - R_{ik,j} \neq 0\) where \(R_{ij}\) is the Ricci tensor, do admit either non-trivial projective transformations, or non-trivial geodesic mappings.
Reviewer: S.Noaghi (Vulcan)

53B20 Local Riemannian geometry