On Killing Vector Fields on Riemannian Manifolds

Sharief Deshmukh and Olga Belova
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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, 2021, vol. 9, issue 3, 1-17

Abstract: We study the influence of a unit Killing vector field on geometry of Riemannian manifolds. For given a unit Killing vector field w on a connected Riemannian manifold ( M , g ) we show that for each non-constant smooth function f ? C ? ( M ) there exists a non-zero vector field w f associated with f . In particular, we show that for an eigenfunction f of the Laplace operator on an n -dimensional compact Riemannian manifold ( M , g ) with an appropriate lower bound on the integral of the Ricci curvature S ( w f , w f ) gives a characterization of the odd-dimensional unit sphere S 2 m + 1 . Also, we show on an n -dimensional compact Riemannian manifold ( M , g ) that if there exists a positive constant c and non-constant smooth function f that is eigenfunction of the Laplace operator with eigenvalue n c and the unit Killing vector field w satisfying ? w 2 ? ( n ? 1 ) c and Ricci curvature in the direction of the vector field ? f ? w is bounded below by n ? 1 c is necessary and sufficient for ( M , g ) to be isometric to the sphere S 2 m + 1 ( c ) . Finally, we show that the presence of a unit Killing vector field w on an n -dimensional Riemannian manifold ( M , g ) with sectional curvatures of plane sections containing w equal to 1 forces dimension n to be odd and that the Riemannian manifold ( M , g ) becomes a K-contact manifold. We also show that if in addition ( M , g ) is complete and the Ricci operator satisfies Codazzi-type equation, then ( M , g ) is an Einstein Sasakian manifold.

Keywords: killing vector field; K-contact manifold; sasakian manifold; Einstein–Sasakian manifold (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2021
References: View complete reference list from CitEc
Citations: View citations in EconPapers (1)

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