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Ricci flow

The Ricci flow (named after [Gregorio Ricci-Curbastro] ) was introduced by [Richard Hamilton] in 1981 in order to gain insight into the [geometrization conjecture] of [William Thurston] , which concerns the [topological classification] of three-dimensional smooth manifolds. Hamilton's idea was to define a kind of nonlinear [diffusion equation] which would tend to smooth out irregularities in the metric. Then, by placing an arbitrary metric g on a given smooth manifold M and evolving the metric by the Ricci flow, the metric should approach a particularly nice metric, which might constitute a [canonical form] for M. Suitable canonical forms had already been identified by Thurston; the possibilities, called Thurston model geometries , include the three-sphere S3, three-dimensional Euclidean space E3, three-dimensional hyperbolic space H3, which are [homogeneous] and [isotropic] , and five slightly more exotic Riemannian manifolds, which are homogeneous but not isotropic. (This list is closely related to, but not identical with, the [Bianchi classification] of the three-dimensional real [Lie algebra] s into nine classes.) Hamilton's idea was that these special metrics should behave like [fixed point] s of the Ricci flow, and that if, for a given manifold, globally only one Thurston geometry was admissible, this might even act like an [attractor] under the flow.

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