Curves in Riemannian Manifolds Making Prescribed Angles With Torse-Forming Vector Fields

TL;DR

This paper introduces prescribed angle curves in Riemannian manifolds related to torse-forming vector fields, establishing existence, curvature formulas, and characterizations of spherical curves.

Contribution

It defines prescribed angle curves with torse-forming vector fields, derives curvature relations, and characterizes spherical curves in space forms.

Findings

Existence of prescribed angle curves with torse-forming vector fields

Explicit curvature formulas in 3D case

Characterization of curves on geodesic spheres

Abstract

In this paper, we introduce the notion of a prescribed angle curve in a Riemannian manifold associated with a pair $(\mathcal{V},\theta)$, where $\mathcal{V}$ is a unit vector field along the curve and $\theta$ denotes the angle between $\mathcal{V}$ and the principal normal vector of the curve. When $\mathcal{V}$ is a torse-forming vector field, we establish an existence result for prescribed angle curves. In the $3$-dimensional case, we determine the curvatures of these curves in terms of the prescribed angle and the potential function of $\mathcal{V}$. Moreover, using this notion, we provide a new characterization of curves lying on geodesic spheres in real space forms.