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

Muhittin Evren Aydin, Esra Dilmen, Busra Karakaya

Abstract

In this paper, we introduce the notion of a prescribed angle curve in a Riemannian manifold associated with a pair $(\mathcal{V},θ)$, where $\mathcal{V}$ is a unit vector field along the curve and $θ$ 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.

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 , where is a unit vector field along the curve and denotes the angle between and the principal normal vector of the curve. When is a torse-forming vector field, we establish an existence result for prescribed angle curves. In the -dimensional case, we determine the curvatures of these curves in terms of the prescribed angle and the potential function of . Moreover, using this notion, we provide a new characterization of curves lying on geodesic spheres in real space forms.

Paper Structure

This paper contains 4 sections, 10 theorems, 45 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. PA curves in Riemannian manifolds
  4. PA curves in Riemannian surfaces

Key Result

Proposition 2.1

cbook Let $M=M_1 \times_{\rho} M_2$ be a warped product, and let $\nabla^0$ and $\nabla^i$ be the Levi-Civita connections on $M$ and $M_i$, respectively. Let also $X_i,Y_i \in \mathcal{L}(M_i)$. Then,

Theorems & Definitions (26)

  • Definition 1.1
  • Proposition 2.1
  • Lemma 3.1
  • Theorem 3.2
  • proof
  • Theorem 3.3
  • proof
  • Remark 3.4
  • Example 3.5
  • Example 3.6
  • ...and 16 more