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

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.