Indian Journal of Science and Technology

Abstract

Objectives: The object of the present paper is to study a type of Riemannian manifold called generalized quasi-conformally 2-recurrent Riemannian manifold, it is denoted by G { e C (2Kn )} . Methods: Differentiable manifold was defined on the basis of topology, differential calculus and real analysis. Riemannian manifold is a part of differentiable manifold in which we study by tensor notation and index free notation. Using such notations, several types of recurrent manifold and their standard results are used to characterize generalized quasi-conformally 2-recurrent Riemannian manifold and also several results of tensor calculus such as contraction, covariant differentiation, divergence of tensor, Bianchi’s identities, Ricci identity, Fundamental theorem of Riemannian geometry and many more are used in this paper to finding the interesting results. Finding: We study G { e C (2Kn )} manifold with Einstein manifold and prove that an Einstein G { e C (2Kn )} is a manifold of constant curvature. We also established a necessary and sufficient condition for an EinsteinG { e C (2Kn )} manifold to be a generalized 2-recurrent Riemannian manifold G { e C (2Kn )} . In the last, we used vanishes Ricci tensor and shows that G { e C (2Kn )} manifold reduces toG { e C (2Kn )} . Novelty: Recurrent manifold and generalized 2- recurrent Riemannian manifold has already been studied in published literature. In this paper we study generalized 2-recurrent Riemannian manifold with quasi-conformal curvature tensor and find interesting results.

Keywords: Recurrent manifold; Ricci tensor; generalized 2-recurrent manifold; quasi-conformal curvature tensor; Einstein manifold