A Note on Killing Calculus on Riemannian Manifolds

Sharief Deshmukh, Amira Ishan, Suha B. Al-Shaikh and Cihan Özgür
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Sharief Deshmukh: Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh 11451, Saudi Arabia
Amira Ishan: Department of Mathematics, College of Science, Taif University, P.O. Box 11099, Taif 21944, Saudi Arabia
Suha B. Al-Shaikh: Information Technology Department, Arab Open University, Hittin P.O. Box 84901, Saudi Arabia
Cihan Özgür: Department of Mathematics, Balıkesir University, Balıkesir 10145, Turkey

Mathematics, 2021, vol. 9, issue 4, 1-13

Abstract: In this article, it has been observed that a unit Killing vector field ? on an n -dimensional Riemannian manifold ( M , g ) , influences its algebra of smooth functions C ? ( M ) . For instance, if h is an eigenfunction of the Laplace operator ? with eigenvalue ? , then ? ( h ) is also eigenfunction with same eigenvalue. Additionally, it has been observed that the Hessian H h ( ? , ? ) of a smooth function h ? C ? ( M ) defines a self adjoint operator ? ? and has properties similar to most of properties of the Laplace operator on a compact Riemannian manifold ( M , g ) . We study several properties of functions associated to the unit Killing vector field ? . Finally, we find characterizations of the odd dimensional sphere using properties of the operator ? ? and the nontrivial solution of Fischer–Marsden differential equation, respectively.

Keywords: Killing vector field; Killing calculus; sphere; isometry (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2021
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