A note on conformal vector fields on a Riemannian manifold

Volume 136 / 2014

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


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.


  • Sharief DeshmukhDepartment of Mathematics
    College of Science
    King Saud University
    P.O. Box 2455
    Riyadh 11451, Saudi Arabia
  • Falleh Al-SolamyDepartment of Mathematics
    Faculty of Science
    King Abdulaziz University
    P.O. Box 80015
    Jeddah 21589, Saudi Arabia

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