Some new characterizations of spheres and Euclidean spaces using conformal vector fields

Given a conformal vector field X defined on an n-dimensional Riemannian manifold (Nn, N n , g ), naturally associated to X are the conformal factor sigma , a smooth function defined on N n , and a skew symmetric (1,1) , 1) tensor field e , 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, N n , g ), n > 1, of positive Ricci curvature admits a nontrivial (non- Killing) conformal vector field X with conformal factor sigma such that its Ricci operator Rc and scalar curvature tau satisfy Rc (X) X ) = - ( n - 1)del sigma del sigma and X ( tau ) = 2 sigma sigma (n(n n ( n - 1)c c - tau ) for a constant c , necessarily c > 0 and (Nn, N n , g ) is isometric to the sphere S n c 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, N n , g ), n > 1, admits a nontrivial conformal vector field X with conformal factor sigma and associated tensor e satisfying Rc (X) X ) = - div e and e (X) X ) = 0, , if and only if (Nn, N n , g ) is isometric to the Euclidean space (En, E n , < , > ).