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On Jacobi-Type Vector Fields on Riemannian Manifolds

Author

Listed:
  • 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)

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.

Suggested Citation

  • Bang-Yen Chen & Sharief Deshmukh & Amira A. Ishan, 2019. "On Jacobi-Type Vector Fields on Riemannian Manifolds," Mathematics, MDPI, vol. 7(12), pages 1-11, November.
    • Handle: RePEc:gam:jmathe:v:7:y:2019:i:12:p:1139-:d:289633
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