Some properties of hyperbolic Yamabe solitons

Abstract

We define the hyperbolic Yamabe flow and obtain some properties of its stationary solutions, namely, of hyperbolic Yamabe solitons. We consider immersed submanifolds as hyperbolic Yamabe solitons and prove that, under certain assumptions, a hyperbolic Yamabe soliton hypersurface is a pseudosymmetric or a metallic shaped hypersurface. We characterize the hyperbolic Yamabe soliton factor manifolds of a multiply twisted, multiply warped, doubly warped, and warped product manifold and provide a classification for a complete gradient hyperbolic Yamabe soliton factor manifold. We also determine the conditions for the factor manifolds to be hyperbolic Yamabe solitons if the manifold is a hyperbolic Yamabe soliton and illustrate this result for a physical model of the universe, namely, for the Robertson--Walker spacetime.