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Open AccessArticle

A Remarkable Property of Concircular Vector Fields on a Riemannian Manifold

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Department of Mathematics and Statistics, College of Science, Imam Mohammad Ibn Saud Islamic University, P.O. Box-65892, Riyadh 11566, Saudi Arabia
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Department of Mathematics, College of Science, King Saud University, P.O. Box-2455, Riyadh 11451, Saudi Arabia
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Institute of Physical and Mathematical Sciences and IT, Immanuel Kant Baltic Federal University, A. Nevsky str. 14, 236016 Kaliningrad, Russia
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Author to whom correspondence should be addressed.
Mathematics 2020, 8(4), 469; https://doi.org/10.3390/math8040469
Received: 6 March 2020 / Revised: 19 March 2020 / Accepted: 20 March 2020 / Published: 29 March 2020
(This article belongs to the Special Issue Riemannian Geometry of Submanifolds)
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. View Full-Text
Keywords: concircular vector field; connecting function; Ricci curvature; isometric to sphere; isometric to Euclidean space concircular vector field; connecting function; Ricci curvature; isometric to sphere; isometric to Euclidean space
MDPI and ACS Style

Al-Dayel, I.; Deshmukh, S.; Belova, O. A Remarkable Property of Concircular Vector Fields on a Riemannian Manifold. Mathematics 2020, 8, 469.

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