We consider an n-dimensional compact Riemannian manifold (M,g) and show that the presence of a non-Killing conformal vector field ξ on M that is also an eigenvector of the Laplacian operator acting on smooth vector fields with eigenvalue λ > 0, together with an upper bound on the energy of the vector field ξ, implies that M is isometric to the n-sphere Sⁿ(λ). We also introduce the notion of φ-analytic conformal vector fields, study their properties, and obtain a characterization of n-spheres using these vector fields.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-cm136-1-7,
author = {Sharief Deshmukh and Falleh Al-Solamy},
title = {A note on conformal vector fields on a Riemannian manifold},
journal = {Colloquium Mathematicae},
volume = {135},
year = {2014},
pages = {65-73},
zbl = {1301.53036},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm136-1-7}
}
Sharief Deshmukh; Falleh Al-Solamy. A note on conformal vector fields on a Riemannian manifold. Colloquium Mathematicae, Tome 135 (2014) pp. 65-73. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm136-1-7/