A Remarkable Property of Concircular Vector Fields on a Riemannian Manifold

Ibrahim Al-Dayel, Sharief Deshmukh and Olga Belova
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Ibrahim Al-Dayel: Department of Mathematics and Statistics, College of Science, Imam Mohammad Ibn Saud Islamic University, P.O. Box-65892, Riyadh 11566, Saudi Arabia
Sharief Deshmukh: Department of Mathematics, College of Science, King Saud University, P.O. Box-2455, Riyadh 11451, Saudi Arabia
Olga Belova: Institute of Physical and Mathematical Sciences and IT, Immanuel Kant Baltic Federal University, A. Nevsky str. 14, 236016 Kaliningrad, Russia

Mathematics, 2020, vol. 8, issue 4, 1-10

Abstract: In this paper, we show that, given a non-trivial concircular vector field u on a Riemannian manifold ( M , g ) with potential function f , there exists a unique smooth function ρ on M that connects u to the gradient of potential function ∇ f . We call the connecting function of the concircular vector field u . This connecting function is shown to be a main ingredient in obtaining characterizations of n -sphere S n ( c ) and the Euclidean space E n . We also show that the connecting function influences on a topology of the Riemannian manifold.

Keywords: concircular vector field; connecting function; Ricci curvature; isometric to sphere; isometric to Euclidean space (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2020
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