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. 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 g 0 on a smooth manifold M is a Ricci soliton if there exists a function σ ( t ) and a family of diffeomorphisms { η ( t ) } ⊂ Diff ⁡ ( M ) such that

is a solution of Ricci flow. In this expression, η ( t ) ∗ g 0 refers to the pullback off the metric g 0 by the diffeomorphism η ( t ) .

Equivalently, a metric g 0 is a Ricci soliton if and only if

where Rc is the Ricci curvature tensor, λ ∈ R , X is a vector field on M , and L represents the Lie derivative. This condition is a generalization of the Einstein condition for metrics: