Ricci Vector Fields

Abstract

We introduce a special vector field ω on a Riemannian manifold (Nm,g), such that the Lie derivative of the metric g with respect to ω is equal to ρRic, where Ric is the Ricci tensor of (Nm,g) and ρ is a smooth function on Nm. We call this vector field a ρ-Ricci vector field. We use the ρ-Ricci vector field on a Riemannian manifold (Nm,g) and find two characterizations of the m-sphere Smα. In the first result, we show that an m-dimensional compact and connected Riemannian manifold (Nm,g) with nonzero scalar curvature admits a ρ-Ricci vector field ω such that ρ is a nonconstant function and the integral of Ricω,ω has a suitable lower bound that is necessary and sufficient for (Nm,g) to be isometric to m-sphere Smα. In the second result, we show that an m-dimensional complete and simply connected Riemannian manifold (Nm,g) of positive scalar curvature admits a ρ-Ricci vector field ω such that ρ is a nontrivial solution of the Fischer–Marsden equation and the squared length of the covariant derivative of ω has an appropriate upper bound, if and only if (Nm,g) is isometric to m-sphere Smα.