A note on conformal vector fields on a Riemannian manifold

Sharief Deshmukh ¹   ; Falleh Al-Solamy ²

¹ Department of Mathematics College of Science King Saud University P.O. Box 2455 Riyadh 11451, Saudi Arabia
² Department of Mathematics Faculty of Science King Abdulaziz University P.O. Box 80015 Jeddah 21589, Saudi Arabia

Colloquium Mathematicum, Tome 136 (2014) no. 1, pp. 65-73

Résumé

We consider an $n$-dimensional compact Riemannian manifold $(M,g)$ and show that the presence of a non-Killing conformal vector field $\xi $ on $M$ that is also an eigenvector of the Laplacian operator acting on smooth vector fields with eigenvalue $\lambda >0 $, together with an upper bound on the energy of the vector field $\xi $, implies that $M$ is isometric to the $n$-sphere $S^{n}(\lambda )$. We also introduce the notion of $\varphi $-analytic conformal vector fields, study their properties, and obtain a characterization of $n$-spheres using these vector fields.

Détail

DOI : 10.4064/cm136-1-7

Keywords: consider n dimensional compact riemannian manifold presence non killing conformal vector field eigenvector laplacian operator acting smooth vector fields eigenvalue lambda together upper bound energy vector field implies isometric n sphere lambda introduce notion varphi analytic conformal vector fields study their properties obtain characterization n spheres using these vector fields