Ricci flow

In differential geometry, the Ricci flow is a process which deforms the metric of a Riemannian manifold in a manner formally analogous to the diffusion of heat.

Relation to Uniformization and Geometrization

The Ricci flow 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 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 well-known Bianchi classification of the three-dimensional real Lie algebras into nine isomorphism classes .) Hamilton's idea was that these special metrics should behave like fixed points 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.

Related Topics:
Richard Hamilton - Geometrization conjecture - William Thurston - Topological classification - Diffusion equation - Canonical form - Bianchi classification - Lie algebra - Fixed points - Attractor

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Hamilton succeeded in proving that any smooth closed three-manifold which admits a metric of positive Ricci curvature also admits a unique Thurston geometry, namely a spherical metric, which does indeed act like an attracting fixed point under the Ricci flow. This doesn't prove the full geometrization conjecture because the most difficult case turns out to concern manifolds with negative Ricci curvature and more specifically those with negative sectional curvature. In this case, mathematicians expect that the Ricci flow should evolve an arbitrary negatively curved three-manifold into one which is locally isometric to H3. Indeed, a triumph of nineteenth century geometry was the proof of the uniformization theorem, the analogous topological classification of smooth two-manifolds, where Hamilton showed that the Ricci flow does indeed evolve a negative curved two-manifold into a two-dimensional multi-holed torus which is locally isometric to the hyperbolic plane. This topic is closely related to important topics in analysis, number theory, dynamical systems, mathematical physics, and even cosmology.

Related Topics:
Locally isometric - Uniformization theorem

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Note that the term "uniformization" correctly suggests a kind of smoothing away of irregularities in the geometry, while the term "geometrization" correctly suggests placing a geometry on a smooth manifold. Geometry is being used here in a precise manner akin to Klein's notion of geometry (see Geometrization conjecture for further details). In particular, the result of geometrization may be a geometry that is not isotropic. In most cases including the cases of constant curvature, the geometry is unique. An important theme in this area is the interplay between real and complex formulations. In particular, many discussions of uniformization speak of complex curves rather than real two-manifolds.

Related Topics:
Klein - Notion of geometry - Geometrization conjecture - Isotropic

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It is possible to construct a kind of superspace of n-dimensional Riemannian manifolds, and then the Ricci flow really does give a flow (in the intuitive sense of particles flowing along flowlines) in this superspace.

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