Conformal metrics and Ricci tensors on the sphere

We consider tensors on the unit sphere , where , is the standard metric and is a differentiable function on . For such tensors, we consider the problems of existence of a Riemannian metric , conformal to , such that , and the existence of such a metric that satisfies , where is the scalar curvature of . We find the restrictions on the Ricci candidate for solvability, and we construct the solutions when they exist. We show that these metrics are unique up to homothety, and we characterize those defined on the whole sphere. As a consequence of these results, we determine the tensors that are rotationally symmetric. Moreover, we obtain the well-known result that a tensor , , has no solution on if and only metrics homothetic to admit as a Ricci tensor. We also show that if , then equation has no solution , conformal to on , and only metrics homothetic to are solutions to this equation when . Infinitely many solutions, globally defined on , are obtained for the equation

where . The geometric interpretation of these solutions is given in terms of existence of complete metrics, globally defined on and conformal to the Euclidean metric, for certain bounded scalar curvature functions that vanish at infinity.