On Concircularly Φ−recurrent Para-sasakian Manifolds

A transformation of an n-dimensional Riemannian manifold M , which transforms every geodesic circle of M into a geodesic circle, is called a concircular transformation. A concircular transformation is always a conformal transformation. Here geodesic circle means a curve in M whose first curvature is constant and second curvature is identically zero. Thus, the geometry of concircular transformations, that is, the concircular geometry, is a generalization of inversive geometry in the sense that the change of metric is more general than that induced by a circle preserving diffeomorphism. An interesting invariant of a concircular transformation is the concircular curavture tensor. The notion of local symmetry of a Riemannian manifold has been studied by many authors ([6], [7]) in several ways and to a different extent. As a weaker version of local symmetry, in 1977, Takahashi [9] introduced the notion of locally φ−symmetric Sasakian manifold and obtained their several interesting results. Later in 2009, De, Yildiz and Yaliniz [5] studied φ−recurrent Kenmotsu manifold and obtained some interesting results too. In this paper we study a concircularly φ−recurrent para-Sasakian manifold which generalizes the notion of locally concircular φ−symmetric para-Sasakian manifold. Again, it is proved that a concircularly φ−recurrent para-Sasakian manifold is an Einstein manifold and in a concircularly φ− recurrent para-Sasakian manifold, the characteristic vector field ξ and the vector field ρ associated to the 1−form A are co-directional. Finally, we proved that a three-dimensional locally concircularly φ−recurrent para-Sasakian manifold is of constant curvature.