Conformal gradient vector fields on a compact Riemannian manifold
Volume 112 / 2008
Colloquium Mathematicum 112 (2008), 157-161
MSC: 53C20, 53A50.
DOI: 10.4064/cm112-1-8
Abstract
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.
