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 $g_0$ on a smooth manifold $M$ is a Ricci soliton if there exists a function $\sigma(t)$ and a family of diffeomorphisms $\{\eta(t)\} \subset \operatorname{Diff}(M)$ such that

$g(t) = \sigma(t) \, \eta(t)^* g_0$

is a solution of Ricci flow. In this expression, $\eta(t)^*g_0$ refers to the pullback off the metric $g_0$ by the diffeomorphism $\eta(t)$.

Equivalently, a metric $g_0$ is a Ricci soliton if and only if

$\operatorname{Rc}(g_0) = \lambda \, g_0 + \mathcal{L}_X g_0,$

where $\operatorname{Rc}$ is the Ricci curvature tensor, $\lambda \in \mathbb{R}$, $X$ is a vector field on $M$, and $\mathcal{L}$ represents the Lie derivative. This condition is a generalization of the Einstein condition for metrics:

$\operatorname{Rc}(g_0) = \lambda \, g_0.$

References

1. ↑ Cao, HUAI-DONG. "RECENT PROGRESS ON RICCI SOLITONS" (PDF). arxiv.org. Arxiv. Retrieved 12 April 2015.

• Chow, Bennet; Knopf, Dan (2004), The Ricci Flow: An Introduction, American Mathematical Society, ISBN 978-0 821-83515-9