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 curvature of (Nm, g) and ρ is a smooth function on N^{m} and call this vector field a ρ-Ricci vector field. We use ρ-Ricci vector field on a Riemannian manifold (Nm, g) and find two characterizations of m-sphere Sm(α). In 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 nonconstant function and the integral of Ric(ω,ω) has a suitable lower bound is necessary and sufficient for (Nm, g) to be isometric to m-sphere Sm(α). In 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 Fischer-Marsden equation and the squared length of the covariant derivative of ω has an appropriate upper bound, if and only if, (Nm, g) to be isometric to m-sphere Sm(α).