Parallel vector fields on the noninvariant hypersurface of a Sasakian manifold

TL;DR

This paper investigates the properties of parallel vector fields on noninvariant hypersurfaces of Sasakian manifolds, establishing conditions under which such hypersurfaces are totally geodesic.

Contribution

It introduces new results on parallel vector fields on noninvariant hypersurfaces of Sasakian manifolds, linking parallelism to the hypersurface being totally geodesic.

Findings

Parallel vector fields imply the hypersurface is totally geodesic.

The study extends understanding of geometric structures on noninvariant hypersurfaces.

Results are established within the framework of $(\,\phi, g, u, v, \lambda)$-structure.

Abstract

In 1970, Samuel I. Goldberg and Kentaro Yano defined the notion of noninvariant hypersurface of a Sasakian manifold [1]. In this paper we have studied the properties of parallel vector fields with respect to induced connection on the noninvariant hypersurface $M$ of a Sasakian manifold $\tilde M$ with $(\phi, g, u, v, \lambda)-$ structure and proved that if the vector field $V$ is parallel with respect to induced connection on $M$ then $M$ is totally geodesic.