Generalised Ricci Solitons

Abstract

We introduce a class of overdetermined systems of partial differential equations of finite type on (pseudo)-Riemannian manifolds that we call the generalised Ricci soliton equations. These equations depend on three real parameters. For special values of the parameters they specialise to various important classes of equations in differential geometry. Among them there are: the Ricci soliton equations, the vacuum near-horizon geometry equations in general relativity, special cases of Einstein-Weyl equations and their projective counterparts, equations for homotheties and Killing's equation. We also prolong the generalised Ricci soliton equations and, by computing differential constraints, we find a number of necessary conditions for a (pseudo)-Riemannian manifold $(M, g)$ to locally admit non-trivial solutions to the generalised Ricci soliton equations in dimensions 2 and 3. The paper provides also a collection of explicit examples of generalised Ricci solitons in dimensions 2 and 3 (in some cases).