Conformal gradient vector fields on a compact Riemannian manifold

Sharief Deshmukh ¹   ; Falleh Al-Solamy ²

¹ Department of Mathematics King Saud University P.O. Box 2455 Riyadh 11451, Saudi Arabia
² Department of Mathematics King Abdul Aziz University P.O. Box 80015 Jeddah 21589, Saudi Arabia

Colloquium Mathematicum, Tome 112 (2008) no. 1, pp. 157-161

Résumé

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.

Détail

DOI : 10.4064/cm112-1-8

Keywords: proved n dimensional compact connected riemannian manifold ricci curvature ric satisfying ric leq n bigg frac lambda bigg constant admits nonzero conformal gradient vector field isometric where lambda first nonzero eigenvalue laplacian operator observed existence nonzero conformal gradient vector field n dimensional compact connected einstein manifold forces have positive scalar curvature ultimately isometric where n scalar curvature manifold