Ricci Vector Fields

We introduce a special vector field ω on a Riemannian manifold ( N m , g ) , such that the Lie derivative of the metric g with respect to ω is equal to ρ R i c , where R i c is the Ricci tensor of ( N m , g ) and ρ is a smooth function on N m . We call this vector field a ρ -Ricci vector field. We use the ρ -Ricci vector field on a Riemannian manifold ( N m , g ) and find two characterizations of the m -sphere S m α . In the first result, we show that an m -dimensional compact and connected Riemannian manifold ( N m , g ) with nonzero scalar curvature admits a ρ -Ricci vector field ω such that ρ is a nonconstant function and the integral of R i c ω , ω has a suitable lower bound that is necessary and sufficient for ( N m , g ) to be isometric to m -sphere S m α . In the second result, we show that an m -dimensional complete and simply connected Riemannian manifold ( N m , 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 ( N m , g ) is isometric to m -sphere S m α .