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## A note on conformal vector fields on a Riemannian manifold

### Tom 136 / 2014

Colloquium Mathematicum 136 (2014), 65-73 MSC: 53C21, 53C24, 53A30. DOI: 10.4064/cm136-1-7

#### Streszczenie

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.

#### Autorzy

• Sharief DeshmukhDepartment of Mathematics
College of Science
King Saud University
P.O. Box 2455
e-mail
• Falleh Al-SolamyDepartment of Mathematics
Faculty of Science
King Abdulaziz University
P.O. Box 80015
Jeddah 21589, Saudi Arabia
e-mail

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