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On some generalized recurrent manifolds. (English) Zbl 1406.53011

Summary: The object of the present paper is to introduce and study a type of non-flat semi-Riemannian manifolds, called, super generalized recurrent manifolds which generalizes both the notion of hyper generalized recurrent manifolds [the first author and A. Patra, Arch. Math., Brno 46, No. 1, 71–78 (2010; Zbl 1240.53035)] and weakly generalized recurrent manifolds [the first two authors, “On weakly generalized recurrent manifolds”, Ann. Univ. Sci. Budap. Rolando Eötvös, Sect. Math. 54, 35–45 (2011)]. The nature of associated 1-forms of a super generalized recurrent manifold is determined and it is proved that on a Roter type manifold [R. Deszcz, “On Roter type manifolds”, in: Proceedings of the 5th conference on geometry and topology of manifolds, 2003. Luxembourg: University of Luxembourg (2003)] such a notion is equivalent to the notion of generalized Ricci-recurrent manifold [U. C. De et al., Tensor, New Ser. 56, No. 3, 312–317 (1995; Zbl 0859.53009)]. We also obtain a sufficient condition for a super generalized recurrent manifold to be a semisymmetric one and the existence of such notion is ensured by a proper example.

MSC:

53B20 Local Riemannian geometry
53B30 Local differential geometry of Lorentz metrics, indefinite metrics
53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.)
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