Hypersurfaces in Riemannian manifolds with torse-forming axes

TL;DR

This paper classifies hypersurfaces in Riemannian manifolds that have a constant inner product with a torse-forming axis vector field, including special cases like constant angle hypersurfaces and torqued vector fields.

Contribution

It provides a classification of hypersurfaces with a constant inner product to a torse-forming axis in Riemannian manifolds, extending known results to new types of vector fields.

Findings

Classification of constant angle hypersurfaces with a torse-forming axis

Results on hypersurfaces with torqued vector fields as axes

Explicit descriptions of hypersurfaces under these conditions

Abstract

In this paper, we study orientable hypersurfaces $N$ in Riemannian manifolds $(M,\langle , \rangle)$ for which the inner product $\langle U, \mathcal{V} \rangle$ is constant, where $U$ is the unit normal vector field to $N$ and $\mathcal{V}$ is a globally defined torse-forming vector field on $M$, called the axis of $N$. When $\mathcal{V}$ is a unit torse-forming vector field, $N$ becomes a constant angle hypersurface with axis $\mathcal{V}$, and we classify such hypersufaces. After that, the case when $\mathcal{V}$ is a torqued vector field is considered and a corresponding classification is given.