On Submanifolds as Riemann Solitons

Abstract

We provide some properties of Riemann solitons with torse-forming potential vector fields, pointing out their relation to Ricci solitons. We also study those Riemann soliton submanifolds isometrically immersed into a Riemannian manifold endowed with a torse-forming vector field, having as potential vector field its tangential component. We consider the minimal and the totally geodesic cases, too, as well as when the ambient manifold is of constant sectional curvature. In particular, we prove that a totally geodesic submanifold isometrically immersed into a Riemannian manifold endowed with a concircular vector field is a Riemann soliton if and only if it is of constant curvature. Furthermore, we show that, if the potential vector field of a minimal hypersurface Riemann soliton isometrically immersed into a Riemannian manifold of constant curvature and endowed with a concircular vector field is of constant length, then it is a metallic shaped hypersurface.