Generalized weakly symmetric Sasakian manifolds

Abstract

‎In this parer‎, ‎we give a necessary condition for Sasakian manifolds to be generalized weakly symmetric‎. ‎We prove the odd-dimensional spheres are the only generalized weakly symmetric Sasakian manifolds‎. ‎Then‎, ‎we show that generalized weakly Ricci-symmetric Sasakian manifolds are Einstein‎. ‎Thereafter‎, ‎we define the sense of weakly parallel Riemannian submanifolds and show that every weakly parallel invariant submanifold of a Sasakian manifold is totally geodesic‎. ‎Finally‎, ‎we provide some examples which verify our main results‎.