On Jacobi-Type Vector Fields on Riemannian Manifolds

Bang-Yen Chen, Sharief Deshmukh and Amira A. Ishan
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Bang-Yen Chen: Department of Mathematics, Michigan State University, 619 Red Cedar Road, East Lansing, MI 48824-1027, USA
Sharief Deshmukh: Department of Mathematics, College of science, King Saud University P.O. Box-2455, Riyadh 11451, Saudi Arabia
Amira A. Ishan: Department of Mathematics, Taif University, Taif 26571, Saudi Arabia

Mathematics, 2019, vol. 7, issue 12, 1-11

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.

Keywords: Jacobi-type vector fields; Killing vector fields; conformal vector fields; Euclidean space (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2019
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