Geodesic Vector Fields on a Riemannian Manifold

Sharief Deshmukh,Nasser Bin Turki,Patrik Peska

doi:10.3390/math8010137

Abstract

A unit geodesic vector field on a Riemannian manifold is a vector field whose integral curves are geodesics, or in other worlds have zero acceleration. A geodesic vector field on a Riemannian manifold is a smooth vector field with acceleration of each of its integral curves is proportional to velocity. In this paper, we show that the presence of a geodesic vector field on a Riemannian manifold influences its geometry. We find characterizations of n-spheres as well as Euclidean spaces using geodesic vector fields.

Highlights

Let ( M, g) be an n-dimensional Riemannian manifold
Where ∇ is the covariant derivative operator with respect to the Riemannian connection on ( M, g) and ρ : M → R is a smooth function called the potential function of the geodesic vector field ξ
Immersions are used to study the geometry of submanifolds. Special vector fields such as unit geodesic vector fields, Killing vector fields, concircular vector fields, conformal vector fields are used in studying geometry as well as topology of a Riemannian manifold

Summary

Introduction

Let ( M, g) be an n-dimensional Riemannian manifold. We call a smooth vector field ξ on M geodesic vector field if. Special vector fields such as unit geodesic vector fields, Killing vector fields, concircular vector fields, conformal vector fields are used in studying geometry as well as topology of a Riemannian manifold (cf [1,2,3,4,6,7,8,9,10,11,16,17,18,19,20,21,22,23,24,25,26,27,28,29]). As observed through above examples, geodesic vector fields have widespread appearance as compared to Killing vector fields and conformal vector fields, which suggests that they may have a role in the geometry of a Riemannian manifolds, and in theory of relativity as well as medical imaging via the Eikonal equation.

Preliminaries

A Characterization of Euclidean Spaces

A Characterization of n-Spheres