Ricci soliton
In differential geometry, a Ricci soliton is a special type of Riemannian metric. Such metrics evolve under Ricci flow only by symmetries of the flow, and they can be viewed as generalizations of Einstein metrics.[1] The concept is named after Gregorio Ricci-Curbastro.
Ricci flow solutions are invariant under diffeomorphisms and scaling, so one is led to consider solutions that evolve exactly in these ways. A metric on a smooth manifold is a Ricci soliton if there exists a function and a family of diffeomorphisms such that
is a solution of Ricci flow. In this expression, refers to the pullback off the metric by the diffeomorphism .
Equivalently, a metric is a Ricci soliton if and only if
where is the Ricci curvature tensor, , is a vector field on , and represents the Lie derivative. This condition is a generalization of the Einstein condition for metrics:
References
- Chow, Bennet; Knopf, Dan (2004), The Ricci Flow: An Introduction, American Mathematical Society, ISBN 978-0 821-83515-9