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Open AccessArticle

On Jacobi-Type Vector Fields on Riemannian Manifolds

1
Department of Mathematics, Michigan State University, 619 Red Cedar Road, East Lansing, MI 48824-1027, USA
2
Department of Mathematics, College of science, King Saud University P.O. Box-2455, Riyadh 11451, Saudi Arabia
3
Department of Mathematics, Taif University, Taif 26571, Saudi Arabia
*
Author to whom correspondence should be addressed.
Mathematics 2019, 7(12), 1139; https://doi.org/10.3390/math7121139
Received: 31 October 2019 / Revised: 15 November 2019 / Accepted: 19 November 2019 / Published: 21 November 2019
(This article belongs to the Special Issue Sasakian Space)
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. View Full-Text
Keywords: Jacobi-type vector fields; Killing vector fields; conformal vector fields; Euclidean space Jacobi-type vector fields; Killing vector fields; conformal vector fields; Euclidean space
MDPI and ACS Style

Chen, B.-Y.; Deshmukh, S.; Ishan, A.A. On Jacobi-Type Vector Fields on Riemannian Manifolds. Mathematics 2019, 7, 1139.

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