Remarks on Scalar Curvature and Concircular Field Equation

Abstract

We show that the scalar curvature of a Riemannian manifold $M$ is constant if it satisfies (i) the concircular field equation and $M$ is compact, (ii) the special concircular field equation. Finally, we show that, if a complete connected Riemannian manifold admits a concircular non-isometric vector field leaving the scalar curvature invariant, and the conformal function is special concircular, then the scalar curvature is a constant.