| Abstrakt: |
Given a conformal vector field X defined on an n -dimensional Riemannian manifold (N n , g) , naturally associated to X are the conformal factor σ , a smooth function defined on N n , 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 (N n , 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 R c and scalar curvature τ satisfy R c (X) = − (n − 1) ∇ σ and X (τ) = 2 σ (n (n − 1) c − τ) for a constant c , necessarily c > 0 and (N n , g) is isometric to the sphere S c n 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 (N n , g) , n > 1 , admits a nontrivial conformal vector field X with conformal factor σ and associated tensor Ω satisfying R c (X) = − d i v Ω and Ω (X) = 0 , if and only if (N n , g) is isometric to the Euclidean space (E n , ⟨ , ⟩) . [ABSTRACT FROM AUTHOR] |
