Ricci soliton


In differential geometry, a complete Riemannian manifold is called a Ricci soliton if, and only if, there exists a smooth vector field such that
for some constant. Here is the Ricci curvature tensor and represents the Lie derivative. If there exists a function such that we call a gradient Ricci soliton and the soliton equation becomes
Note that when or the above equations reduce to the Einstein equation. For this reason Ricci solitons are a generalization of Einstein manifolds.

Self-similar solutions to Ricci flow

A Ricci soliton yields a self-similar solution to the Ricci flow equation
In particular, letting
and integrating the time-dependent vector field to give a family of diffeormorphisms, with the identity, yields a Ricci flow solution by taking
In this expression refers to the pullback of the metric by the diffeomorphism. Therefore, up to diffeomorphism and depending on the sign of, a Ricci soliton homothetically shrinks, remains steady or expands under Ricci flow.

Examples of Ricci solitons

Shrinking ( \lambda > 0 )

Shrinking and steady Ricci solitons are fundamental objects in the study of Ricci flow as they appear as blow-up limits of singularities. In particular, it is known that all Type I singularities are modeled on non-collapsed gradient shrinking Ricci solitons. Type II singularities are expected to be modeled on steady Ricci solitons in general, however to date this has not been proven, even though all known examples are.