Narrow Results

We consider a nearly hyperbolic Sasakian manifold equipped with $(f,g,u,v,\lambda )$-structure and study non-invariant hypersurface of a nearly hyperbolic Sasakian manifold equipped with $(f,g,u,v,\lambda )$-structure. We obtain some properties of nearly hyperbolic Sasakian manifold equipped with $(f,g,u,v,\lambda )$-structure. Further, we find the necessary and sufficient conditions for totally umbilical non-invariant hypersurface with $(f,g,u,v,\lambda )$-structure of nearly hyperbolic Sasakian manifold to be totally geodesic. We also calculate the second fundamental form of a non-invariant hypersurface of a nearly hyperbolic Sasakian manifold with $(f,g,u,v,\lambda )$-structure under the condition when f is parallel.

In this paper, complete spacelike submanifolds with parallel normalized mean curvature vector are investigated in semi-Riemannian space obeying some standard curvature conditions. In this setting, we obtain a suitable Simons type formula and apply it jointly with the well-known generalized maximum principle of Omori–Yau to show that it must be totally umbilical submanifold or isometric to an isoparametric hypersurface in a submanifold $L_1^{n+1}$ of $L_p^{n+p}$.

This is a survey of the principal results about the geodesic completeness of nondegenerate hypersurfaces in Lorentzian manifolds from a structural point of view. Some of these results retain their validity in the case of semi-Riemannian submanifolds in semi-Euclidean spaces, as well.

The object of this paper is to study anti-invariant submanifolds of trans-Sasakian manifolds. We characterize such submanifolds on the basis of parallelism, semi-parallelism and pseudo parallelism of the second fundamental form of the submanifolds. We also characterize totally umbilical anti-invariant submanifolds of trans-Sasakian manifolds. Existence of Legendre curves on such submanifolds has been analyzed. Whether an anti-invariant submanifold of a trans-Sasakian manifold inherits local symmetry from ambient space is investigated here. Nature of Ricci soliton on anti-invariant submanifolds of trans-Sasakian manifolds has been characterized.