On Jacobi-Type Vector Fields on Riemannian Manifolds

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 tha...

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Bibliographic Details
Main Authors: Bang-Yen Chen, Sharief Deshmukh, Amira A. Ishan
Format: Article
Language:English
Published: MDPI AG 2019-11-01
Series:Mathematics
Subjects:
Online Access:https://www.mdpi.com/2227-7390/7/12/1139
Description
Summary: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.
ISSN:2227-7390