Conformal gradient vector fields on a compact Riemannian manifold

Volume 112 / 2008

Sharief Deshmukh, Falleh Al-Solamy Colloquium Mathematicum 112 (2008), 157-161 MSC: 53C20, 53A50. DOI: 10.4064/cm112-1-8


It is proved that if an $n$-dimensional compact connected Riemannian manifold $(M,g)$ with Ricci curvature ${\rm Ric}$ satisfying $$ 0<{\rm Ric}\leq (n-1)\bigg( 2-\frac{nc}{\lambda _{1}}\bigg) c $$ for a constant $c$ admits a nonzero conformal gradient vector field, then it is isometric to $S^{n}(c)$, where $\lambda _{1}$ is the first nonzero eigenvalue of the Laplacian operator on $M$. Also, it is observed that existence of a nonzero conformal gradient vector field on an $n$-dimensional compact connected Einstein manifold forces it to have positive scalar curvature and ultimately to be isometric to $S^{n}(c)$, where $n(n-1)c$ is the scalar curvature of the manifold.


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

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