On some generalized recurrent manifolds

ýThe object of the present paper is to introduce and study a type of non-flat semi-Riemannian manifoldsý, ýcalledý, ýsuper generalized recurrent manifolds which generalizes both the notion of hyper generalized recurrent manifolds [ýA.Aý. ýShaikh and Aý. ýPatraý, On a generalized class of recurrent manifoldsý, Archý. ýMathý. ý(Brno) 46 (2010) 71--78ý.] and weakly generalized recurrent manifolds [ýA.Aý. ýShaikh and Iý. ýRoyý, On weakly generalized recurrent manifoldsý, Anný. ýUnivý. ýSciý. ýBudapest Rolando Eotvosý, ýSectý. ýMath. ý54 (2011) 35--45ý.]ý. ýThe nature of associated 1-forms of a super generalized recurrent manifold is determined and it is proved that on a Roter type manifold [ýRý. ýDeszczý, On Roter type manifoldsý,ý iný: ý5th Conference on Geometry and Topology of Manifoldsý, ýKrynicaý, ýPolandý, ý2003ý.] such a notion is equivalent to the notion of generalized Ricci-recurrent manifold [ýU.Cý. ýDeý, ýNý. ýGuha and Dý. ýKamilyaý, On generalized Ricci-recurrent manifoldsý, Tensor (N.S.) 56 (1995)ý, ýnoý. ý3ý, ý312--317ý.]ý. ýWe 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.