It is proved that if an n-dimensional compact connected Riemannian manifold (M,g) with Ricci curvature Ric satisfying 0 < Ric ≤ (n-1)(2-nc/λ₁)c for a constant c admits a nonzero conformal gradient vector field, then it is isometric to Sⁿ(c), where λ₁ 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 Sⁿ(c), where n(n-1)c is the scalar curvature of the manifold.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-cm112-1-8,
author = {Sharief Deshmukh and Falleh Al-Solamy},
title = {Conformal gradient vector fields on a compact Riemannian manifold},
journal = {Colloquium Mathematicae},
volume = {111},
year = {2008},
pages = {157-161},
zbl = {1135.53022},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm112-1-8}
}
Sharief Deshmukh; Falleh Al-Solamy. Conformal gradient vector fields on a compact Riemannian manifold. Colloquium Mathematicae, Tome 111 (2008) pp. 157-161. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm112-1-8/