On Jacobi-Type Vector Fields on Riemannian Manifolds

Bang-Yen Chen,Sharief Deshmukh,Amira A Ishan

doi:10.3390/math7121139

Bang-Yen Chen, Sharief Deshmukh + Show 1 more

Open Access

https://doi.org/10.3390/math7121139

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Abstract

In this article, we study Jacobi-type vector fields on Riemannian manifolds. A Killing vector field is a Jacobi-type vector field while the converse is not true, leading to a natural question of finding conditions under which a Jacobi-type vector field is Killing. In this article, we first prove that every Jacobi-type vector field on a compact Riemannian manifold is Killing. Then, we find several necessary and sufficient conditions for a Jacobi-type vector field to be a Killing vector field on non-compact Riemannian manifolds. Further, we derive some characterizations of Euclidean spaces in terms of Jacobi-type vector fields. Finally, we provide examples of Jacobi-type vector fields on non-compact Riemannian manifolds, which are non-Killing.

Highlights

Throughout this article, we assume that manifolds are connected and differentiable
The presence of a non-zero Killing vector field on a compact Riemannian manifold constrains its geometry, as well as topology; for instance, it does not allow the Riemannian manifold to have negative Ricci curvature, and on a Riemannian manifold of positive curvature, its fundamental group contains a cyclic subgroup with a constant index depending only on n
In Riemannian geometry, Jacobi vector fields are vector fields along a geodesic defined by the Jacobi equation that arise naturally in the study of the exponential map

Summary

Introduction

Throughout this article, we assume that manifolds are connected and differentiable. There are several important types of smooth vector fields on an n-dimensional Riemannian manifold ( M, g), whose existence influences the geometry of the Riemannian manifold M. A vector field J along a geodesic γ in a Riemannian manifold M is called a Jacobi vector field if it satisfies the Jacobi equation (cf [7]): D2. A vector field η on a Riemannian manifold M is called a Jacobi-type vector field if it satisfies the following Jacobi-type equation:. The second interesting question is Question 2: “Under what conditions is a Jacobi-type vector field on a non-compact Riemannian manifold a Killing vector field?”. We provide some explicit examples of non-Killing Jacobi-type vector fields

Preliminaries

Jacobi-Type Vector Fields on Compact Riemannian Manifolds

Jacobi-Type Vector Fields on Non-Compact Riemannian Manifolds

Jacobi-Type Vector Fields on Euclidean Spaces

Examples of Non-Killing Jacobi-Type Vector Fields