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A Remarkable Property of Concircular Vector Fields on a Riemannian Manifold

Author

Listed:
  • 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)

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.

Suggested Citation

  • Ibrahim Al-Dayel & Sharief Deshmukh & Olga Belova, 2020. "A Remarkable Property of Concircular Vector Fields on a Riemannian Manifold," Mathematics, MDPI, vol. 8(4), pages 1-10, March.
    • Handle: RePEc:gam:jmathe:v:8:y:2020:i:4:p:469-:d:338370
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