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 of 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

↑ Cao, HUAI-DONG. "RECENT PROGRESS ON RICCI SOLITONS". arXiv:0908.2006.

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

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[Cao2009-1] Cao, HUAI-DONG. "RECENT PROGRESS ON RICCI SOLITONS". arXiv:0908.2006.