Ricci solitons on general relativistic spacetimes

Abstract

The main aim of this manuscript is to characterize the general relativistic spacetimes with Ricci and gradient Ricci solitons. It is proven that if the metric of a general relativistic spacetime (M 4, ξ) admitting a special unit timelike vector field ξ is an almost Ricci soliton (g, ξ, λ), then (M 4, ξ) is a perfect fluid spacetime, and almost Ricci soliton (g, ξ, λ) on (M 4, ξ) becomes shrinking Ricci soliton. We prove that a general relativistic perfect fluid spacetime equipped with a special unit timelike vector field together with a Ricci soliton is an Einstein spacetime. In this sequel, we also prove that the Ricci soliton is shrinking, soliton vector field is Killing and the scalar curvature of the perfect fluid spacetime is constant. It is proven that a general relativistic perfect fluid spacetime together with a Ricci soliton is a generalized Robertson-Walker (GRW) spacetime. The existence of gradient Ricci solitons on general relativistic spacetimes are established. We also construct a non-trivial example of general relativistic spacetime equipped with a special unit timelike vector field, and verify some of our theorems.