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. In mathematics, differential topology is the field dealing with differentiable functions on differentiable manifolds. ... In Riemannian geometry, a Riemannian manifold is a real differentiable manifold in which each tangent space is equipped with an inner product in a manner which varies smoothly from point to point. ...

Mathematical definition

Given a Riemannian manifold with metric tensor g_ab, we can compute the Ricci tensor R_ab, which collects averages of sectional curvatures into a kind of "trace" of the Riemann curvature tensor. If we consider the metric tensor (and the associated Ricci tensor) to be functions of a variable which is usually called "time" (but which may have nothing to do with any physical time), then the Ricci flow may be defined by the evolution equation In mathematics, in Riemannian geometry, the metric tensor is a tensor of rank 2 that is used to measure distance and angle in a space. ... In differential geometry, the Ricci curvature tensor is (0,2)-valent tensor, obtained as a trace of the full curvature tensor. ... In differential geometry, the Riemann curvature tensor is the most standard way to express curvature of Riemannian manifolds, or more generally, any manifold with an affine connection, torsionless or with torsion. ...

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 constant curvature metric, which might constitute a canonical form for M. Suitable canonical forms had already been identified by Thurston; the possibilities 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. Richard Hamilton can refer to one of several people: Richard Hamilton, a British painter and collage artist; Richard Hamilton, a basketball player with the Detroit Pistons of the National Basketball Association; and Richard S. Hamilton, Professor of Mathematics at Columbia University. ... The geometrization conjecture, also known as Thurstons geometrization conjecture, concerns the geometric structure of compact 3-manifolds. ... William Paul Thurston (born October 30, 1946) is an American mathematician. ... In the mathematical field of topology a homeomorphism or topological isomorphism (from the Greek words homeos = identical and morphe = shape) is a special isomorphism between topological spaces which respects topological properties. ... The heat equation or diffusion equation is an important partial differential equation which describes the variation of temperature in a given region over time. ... Generally, in mathematics, a canonical form is a function that is written in the most standard, conventional, and logical way. ... In mathematics, a Lie algebra is an algebraic structure whose main use lies in studying geometric objects such as Lie groups and differentiable manifolds. ... In mathematics, a fixed point of a function f is an argument x such that f(x) = x; see fixed point (mathematics). ... In dynamical systems, an attractor is a set to which the system evolves after a long enough time. ...

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. In geometry and mathematical analysis, an isometry is a bijective distance-preserving mapping. ... In mathematics, the uniformization theorem for surfaces says that any surface admits a Riemannian metric of constant Gauss curvature. ...

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 unique geometry on a smooth manifold, which serves as a canonical form and also yields the desired classification. 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.

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. Superspace has had two meanings in physics. ...

Relation to diffusion

To see why the evolution equation defining the Ricci flow is indeed a kind of nonlinear diffusion equation, we can consider the special case of (real) two-manifolds in more detail. Any metric tensor on a two-manifold can be written with respect to an exponential isothermal coordinate chart in the form

(These coordinates provide an example of a conformal coordinate chart, because angles, but not distances, are correctly represented.) In mathematics, a conformal map is a function which preserves angles. ...

The easiest way to compute the Ricci tensor and Laplace-Beltrami operator for our Riemannian two-manifold is to use the differential forms method of Élie Cartan. Take the coframe field In differential geometry, the Ricci curvature tensor is (0,2)-valent tensor, obtained as a trace of the full curvature tensor. ... In mathematics and physics, the Laplace operator or Laplacian, denoted by Δ, is a differential operator, specifically an important case of an elliptic operator (or a hyperbolic operator when defined on pseudo-Riemannian manifolds), with many applications in mathematics and physics. ... Élie Joseph Cartan (9 April 1869 - 6 May 1951) was an influential French mathematician, who did fundamental work in the theory of Lie groups and their geometric applications. ...

so that metric tensor becomes In mathematics, in Riemannian geometry, the metric tensor is a tensor of rank 2 that is used to measure distance and angle in a space. ...

Next, given an arbitrary smooth function h(x,y), compute the exterior derivative In mathematics, the exterior derivative operator of differential topology, extends the concept of the differential of a function to differential forms of higher degree. ...

Take the Hodge dual In mathematics, the Hodge star operator is a linear map on the exterior algebra of an oriented inner product space which establishes a correspondence between the space of k-vectors and the space of (n-k)-vectors. ...

Take another exterior derivative

(where we used the anti-commutative property of the exterior product). That is, In mathematics, the exterior algebra (also known as the Grassmann algebra) of a given vector space V is a certain unital associative algebra which contains V as a subspace. ...

Taking another Hodge dual gives

which gives the desired expression for the Laplace/Beltrami operator

To compute the curvature tensor, we take the exterior derivative of the covector fields making up our coframe:

From these expressions, we can read off the only independent connection one-form

Take another exterior derivative

This gives the curvature two-form

from which we can read off the only linearly independent component of the Riemann tensor using In differential geometry, the Riemann curvature tensor is the most standard way to express curvature of Riemannian manifolds, or more generally, any manifold with an affine connection, torsionless or with torsion. ...

Namely

from which the only nonzero components of the Ricci tensor are In differential geometry, the Ricci curvature tensor is (0,2)-valent tensor, obtained as a trace of the full curvature tensor. ...

R₂₂ = R₁₁ = − Δp

From this, we find components with respect to the coordinate cobasis, namely

But the metric tensor is also diagonal, with

g_xx = g_yy = exp(2p)

and after some elementary manipulation, we obtain an elegant expression for the Ricci flow:

This is manifestly analogous to the best known of all diffusion equations, the heat equation The heat equation or diffusion equation is an important partial differential equation which describes the variation of temperature in a given region over time. ...

where now is the usual Laplacian on the Euclidean plane. The reader may object that the heat equation is of course a linear partial differential equation--- where is the promised nonlinearity in the p.d.e. defining the Ricci flow? In vector calculus, the Laplace operator or Laplacian is a differential operator equal to the sum of all the unmixed second partial derivatives of a dependent variable. ... The word linear comes from the Latin word linearis, which means created by lines. ... In mathematics, and in particular analysis, a partial differential equation (PDE) is an equation involving partial derivatives of an unknown function. ...

The answer is that nonlinearity enters because the Laplace-Beltrami operator depends upon the same function p which we used to define the metric. But notice that the flat Euclidean plane is given by taking p(x,y) = 0. So if p is small in magnitude, we can consider it to define small deviations from the geometry of a flat plane, and if we retain only first order terms in computing the exponential, the Ricci flow on our two-dimensional almost flat Riemannian manifold becomes the usual two dimensional heat equation. This computation suggests that, just as (according to the heat equation) an irregular temperature distribution in a hot plate tends to become more homogeneous over time, so too (according to the Ricci flow) an almost flat Riemannian manifold will tend to flatten out the same way that heat can be carried off "to infinity" in an infinite flat plate. But if our hot plate is finite in size, and has no boundary where heat can be carried off, we can expect to homogenize the temperature, but clearly we cannot expect to reduce it to zero. In the same way, we expect that the Ricci flow, applied to a distorted round sphere, will tend to round out the geometry over time, but not to turn it into a flat Euclidean geometry.

Recent developments

The Ricci flow has been intensively studied since 1981. Some recent work has focused on the question of precisely how higher dimensional Riemannian manifolds evolve under the Ricci flow, and in particular, what types of parametric singularities may form. For instance, a certain class of solutions to the Ricci flow demonstrates that neckpinch singularities will form on an evolving n-dimensional metric Riemannian manifold having a certain topological property (positive Euler characteristic), as the flow approaches some characteristic time t₀. In certain cases such neckpinches will produce manifolds called Ricci solitons. Singularity has several different meanings: mathematical singularity - a point where a mathematical function goes to infinity or is in certain other ways ill-behaved gravitational singularity - an infinity occurring in an astrophysical model, involving infinite curvature (a mathematical singularity) in the space/time continuum technological singularity - a predicted point in... In algebraic topology, the Euler characteristic is a topological invariant (in fact, homotopy invariant) defined for a broad class of topological spaces. ...

Many variants of the Ricci flow have also been studied:

• Various curvature flows defined using either an extrinsic curvature, which describes how a curve or surface is embedded in a higher dimensional flat space, or an intrinsic curvature, which describes the internal geometry of some Riemannian manifold,
• Various flows which extremalize some quantity mathematically analogous to an energy or entropy,
• Various flows controlled by a p.d.e. which is a higher order analog of a nonlinear diffusion equation.

Some of the most interesting variants are examples of all of these possibilities. In particular, the Calabi flow, which, like the Ricci flow, is an intrinsic curvature flow. This flow tends to smooth out deviations from roundness in a manner formally analogous to the way that the two-dimensional vibration equation damps and propagates away transverse mechanical vibrations in a thin plate, and it extremalizes a certain intrinsic curvature functional. The Calabi flow is important in the study of Calabi-Yau manifolds and also in the study of Robinson/Trautman spacetimes in general relativity. An intriguing observation is that the underlying Calabi equation appears to be completely integrable, which would give a direct link with the theory of solitons. In mathematics, see Embedding. ... In Riemannian geometry, a Riemannian manifold is a real differentiable manifold in which each tangent space is equipped with an inner product in a manner which varies smoothly from point to point. ... In differential geometry, the Calabi flow is a process which deforms the metric of a Riemannian manifold (or better yet, a Kähler manifold) in a manner formally analogous to the way that vibrations are damped and dissipated in a hypothetical curved n-dimensional structural element. ... In mathematics, a Calabi-Yau manifold is a compact K hler manifold with a vanishing first Chern class. ... Two-dimensional visualization of space-time distortion. ... A soliton is a self-reinforcing solitary wave caused by nonlinear effects in the medium. ...

Curvature flows may or may not preserve volume. The Calabi flow does; the Ricci flow does not, so to be more careful in applying the Ricci flow to uniformization we'd need to normalize the Ricci flow to obtain a flow which preserves volume. If we fail to do this, the problem is that (for example) instead of evolving a given three-dimensional manifold into one Thurston's canonical forms, we might just shrink its size.

See also

• uniformization theorem,
• geometrization conjecture,
• Fuchsian group,
• automorphic function,
• Falting's theorem,
• calculus of variations,
• Kähler manifold.

In mathematics, the uniformization theorem for surfaces says that any surface admits a Riemannian metric of constant Gauss curvature. ... The geometrization conjecture, also known as Thurstons geometrization conjecture, concerns the geometric structure of compact 3-manifolds. ... In mathematics, a Fuchsian group is a particular type of group of isometries of the hyperbolic plane. ... In mathematics, the general notion of automorphic form is the extension to analytic functions, perhaps of several complex variables, of the theory of modular forms. ... In number theory, the Mordell conjecture stated a basic result regarding the rational number solutions to Diophantine equations. ... Calculus of variations is a field of mathematics which deals with functions of functions, as opposed to ordinary calculus which deals with functions of numbers. ... In mathematics, a Kähler manifold is a complex manifold which also carries a Riemannian metric and a symplectic form on the underlying real manifold in such a way that the three structures (complex, Riemannian, and symplectic) are all mutually compatible. ...

References

• Bakas, I.. "The algebraic structure of geometric flows in two dimensions." arXiv eprint server. Accessed on July 28, 2005.

• Chow, Bennet and Knopf, Dan (2004). The Ricci Flow: an introduction. American Mathematical Society. ISBN 0821835157..

• Weeks, Jeffery R. (1985). The Shape of Space: how to visualize surfaces and three-dimensional manifolds. New York: Marcel Dekker. ISBN 0-824-77437-X.. A superb popular book which aims to explain the background for the Thurston classification programme.