Pseudoparallel Invariant Submanifolds of Lorentzian α-sasakian Manifolds

Abstract

In this paper, the study of an invariant submanifold of Lorentzian α- sasakian manifold is carried out and it is shown that, it is also Lorentzian α-sasakian. Further we prove that, if the second fundamental form of an invariant submanifold of Lorentzian α-sasakian manifold is recurrent, 2-recurrent and generalized 2-recurrent then the submanifold is totally geodesic and also an invariant submanifold of Lorentzian α-sasakian manifold with parallel third fundamental form is again totally geodesic. It is proved that pseudoparallel and 2-pseudoparallel invariant submanifolds of Lorentzian α-sasakian manifolds is also totally geodesic. Further, we also show that this property of totally geodesic holds true if e C · σ = L1Q(g, σ) and e C · e∇ σ = L1Q(g,e∇ σ).