On some generalized recurrent manifolds

Abstract

‎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 [‎A.A‎. ‎Shaikh and A‎. ‎Patra‎, On a generalized class of recurrent manifolds‎, Arch‎. ‎Math‎. ‎(Brno) 46 (2010) 71--78‎.] and weakly generalized recurrent manifolds [‎A.A‎. ‎Shaikh and I‎. ‎Roy‎, On weakly generalized recurrent manifolds‎, Ann‎. ‎Univ‎. ‎Sci‎. ‎Budapest Rolando Eotvos‎, ‎Sect‎. ‎Math. ‎54 (2011) 35--45‎.]‎. ‎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: ‎5th Conference on Geometry and Topology of Manifolds‎, ‎Krynica‎, ‎Poland‎, ‎2003‎.] such a notion is equivalent to the notion of generalized Ricci-recurrent manifold [‎U.C‎. ‎De‎, ‎N‎. ‎Guha and D‎. ‎Kamilya‎, On generalized Ricci-recurrent manifolds‎, Tensor (N.S.) 56 (1995)‎, ‎no‎. ‎3‎, ‎312--317‎.]‎. ‎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.