Generalised Ricci Solitons

TL;DR

This paper introduces a broad class of overdetermined PDE systems called generalised Ricci soliton equations on (pseudo)-Riemannian manifolds, unifying various important geometric equations and providing conditions and examples for their solutions.

Contribution

It defines the generalised Ricci soliton equations depending on parameters, relates them to known geometric equations, and finds necessary conditions and explicit examples in low dimensions.

Findings

Derived necessary conditions for solutions in dimensions 2 and 3.

Prolonged the equations and computed differential constraints.

Provided explicit examples of solutions in low dimensions.

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).