Prescribed Angle Surfaces Associated with Torse-Forming Vector Fields in Riemannian Manifolds

TL;DR

This paper introduces and classifies prescribed angle hypersurfaces in Riemannian manifolds associated with torse-forming vector fields, analyzing their curvature properties in three dimensions.

Contribution

It defines prescribed angle hypersurfaces linked to torse-forming vector fields and provides a classification and curvature analysis in three-dimensional cases.

Findings

Determined intrinsic and extrinsic curvatures of prescribed angle hypersurfaces.

Classified prescribed angle surfaces under certain conditions.

Expressed curvatures in terms of angle and potential function.

Abstract

In this paper, we introduce the notion of a prescribed angle hypersurface in a Riemannian manifold associated with a pair $(\mathcal{V},\theta)$, where $\mathcal{V}$ is a unit vector field along the hypersurface and $\theta$ denotes the angle between $\mathcal{V}$ and the unit normal vector field of the hypersurface. We study such hypersurfaces in case $\mathcal{V}$ is a torse-forming vector field. In the particular $3$-dimensional case, we determine the intrinsic and extrinsic curvatures of these hypersurfaces in terms of the prescribed angle and the potential function of $\mathcal{V}$. Using this, we classify prescribed angle surfaces under suitable assumptions.