@article{Deshmukh2024, 
author = {Sharief Deshmukh and Mohammed Guediri},
title = {Some new characterizations of spheres and Euclidean spaces using conformal vector fields},
year = {2024},
journal = {AIMS Mathematics},
volume = {9},
number = {10},
pages = {28765-28777},
keywords = {conformal field, conformal factor, isometric to sphere, isometric to Euclidean space},
url = {https://www.sciopen.com/article/10.3934/math.20241395},
doi = {10.3934/math.20241395},
abstract = {Given a conformal vector field  X defined on an  n-dimensional Riemannian manifold  (Nn,g), naturally associated to  X are the conformal factor  σ, a smooth function defined on  Nn, and a skew symmetric  (1,1) tensor field  Ω, called the associated tensor, that is defined using the  1-form dual to  X. In this article, we prove two results. In the first result, we show that if an  n-dimensional compact and connected Riemannian manifold  (Nn,g),  n>1, of positive Ricci curvature admits a nontrivial (non-Killing) conformal vector field  X with conformal factor  σ such that its Ricci operator  Rc and scalar curvature  τ satisfy   Rc(X)=−(n−1)∇σandX(τ)=2σ(n(n−1)c−τ)for a constant  c, necessarily  c>0 and  (Nn,g) is isometric to the sphere  Scn of constant curvature  c. The converse is also shown to be true. In the second result, it is shown that an  n-dimensional complete and connected Riemannian manifold  (Nn,g),  n>1, admits a nontrivial conformal vector field  X with conformal factor  σ and associated tensor  Ω satisfying   Rc(X)=−divΩandΩ(X)=0,if and only if  (Nn,g) is isometric to the Euclidean space  (En,⟨,⟩).}
}