Generalized Weakly Symmetric Sasakian Manifolds

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.