Ricci flow - Citizendia

In differential geometry, the Ricci flow is an intrinsic geometric flow—a process which deforms the metric of a Riemannian manifold—in this case in a manner formally analogous to the diffusion of heat, thereby smoothing out irregularities in the metric. Differential geometry is a mathematical discipline that uses the methods of differential and integral Calculus to study problems in Geometry In Mathematics, specifically Differential geometry, a geometric flow is the Gradient flow associated to a functional on a Manifold which has In Riemannian geometry, a Riemannian manifold ( M, g) (with Riemannian metric g) is a real Differentiable manifold M It plays an important role in the proof of the Poincaré conjecture, one of the seven Millennium Prize Problems for which the Clay Mathematics Institute offers a $1,000,000 prize for a correct solution; see the Solution of the Poincaré conjecture, and in this context is also called the Ricci-Hamilton flow. In Mathematics, the Poincaré conjecture (French pwɛ̃kaʀe is a Theorem about the characterization of the three-dimensional sphere among The Millennium Prize Problems are seven problems in Mathematics that were stated by the Clay Mathematics Institute in 2000 The Clay Mathematics Institute (CMI is a private Non-profit foundation, based in Cambridge, Massachusetts. This entry describes the solution of the Poincaré conjecture at a level intended for the general public

1 Mathematical definition
2 Examples
3 Relationship to uniformization and geometrization
4 Relation to diffusion
5 Recent developments
6 See also
- 6.1 Applications
- 6.2 General context
7 References

Mathematical definition

Given a Riemannian manifold with metric tensor $g i j$ , we can compute the Ricci tensor $R i j$ , which collects averages of sectional curvatures into a kind of "trace" of the Riemann curvature tensor. In the mathematical field of Differential geometry, a metric tensor is a type of function defined on a Manifold (such as a Surface in space In Differential geometry, the Ricci curvature tensor, named after Gregorio Ricci-Curbastro, provides one way of measuring the degree to which the geometry determined In the Mathematical field of Differential geometry, the Riemann curvature tensor or Riemann–Christoffel tensor is the most standard way to express 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 geometric evolution equation

$\partial_t g_{ij}=-2 R_{ij}.$

The normalized Ricci flow makes sense for compact manifolds and is given by the equation

$\partial_t g_{ij}=-2 R_{ij} +\frac{2}{n} R_\mathrm{avg} g_{ij}$

where $R avg$ is the average (mean) of the scalar curvature (which is obtained from the Ricci tensor by taking the trace) and $n$ is the dimension of the manifold. This normalized equation preserves the volume of the metric.

The factor of −2 is of little significance, since it can be changed to any nonzero real number by rescaling t. However the minus sign ensures that the Ricci flow is well defined for sufficiently small positive times; if the sign is changed then the Ricci flow would usually only be defined for small negative times. (This is similar to the way in which the heat equation can be run forwards in time, but not usually backwards in time. )

Informally, the Ricci flow tends to expand negatively curved regions of the manifold, and contract positively curved regions.

Examples

If the manifold is Euclidean space, or more generally Ricci-flat, then Ricci flow leaves the metric unchanged. In Mathematics, Ricci-flat manifolds are Riemannian manifolds whose Ricci curvature vanishes Conversely, any metric unchanged by Ricci flow is Ricci-flat. In Mathematics, Ricci-flat manifolds are Riemannian manifolds whose Ricci curvature vanishes
If the manifold is a sphere (with the usual metric) then Ricci flow collapses the manifold to a point in finite time. If the sphere has radius 1 in n dimensions, then after time t the metric will be multiplied by (1−2t(n-1)), so the manifold will collapse after time 1/2(n−1). More generally, if the manifold is an Einstein manifold (Ricci = constant×metric), then Ricci flow will collapse it to a point if it has positive curvature, leave it invariant if it has zero curvature, and expand it if it has negative curvature. In Differential geometry and Mathematical physics, an Einstein manifold is a Riemannian or Pseudo-Riemannian manifold whose Ricci tensor
For a compact Einstein manifold, the metric is unchanged under normalized Ricci flow. In Differential geometry and Mathematical physics, an Einstein manifold is a Riemannian or Pseudo-Riemannian manifold whose Ricci tensor Conversely, any metric unchanged by normalized Ricci flow is Einstein.

In particular, this shows that in general the Ricci flow cannot be continued for all time, but will produce singularities. For 3 dimensional manifold, Perelman showed how to continue past the singularities using surgery on the manifold.

A significant 2-dimensional example is the cigar soliton solution which is given by the metric (dx²+dy²)/(e^4t+x²+y²) on the Euclidean plane. Although this metric shrinks under the Ricci flow, its geometry remains the same. Such solutions are called steady Ricci solitons. An example of a 3-dimensional steady Ricci soliton is the "Bryant soliton", which is rotationally symmetric, has positive curvature, and is obtained by solving a system of ordinary differential equations.

Relationship 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. Richard Streit Hamilton (born 1943 is professor of Mathematics at Columbia University. Thurston's geometrization conjecture states that compact 3-manifolds can be decomposed into Submanifolds that have geometric structures William Paul Thurston (born October 30, 1946) is an American Mathematician. Topological equivalence redirects here see also Topological equivalence (dynamical systems. Hamilton's idea was to define a kind of nonlinear diffusion equation which would tend to smooth out irregularities in the metric. The heat equation is an important Partial differential equation which describes the distribution of Heat (or variation in temperature in a given region over time 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. Generally in Mathematics, a canonical form (often called normal form or standard form) of an object is a standard way of presenting that object 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 homogenous and isotropic, and five slightly more exotic Riemannian manifolds, which are homogeneous but not isotropic. Isotropy is uniformity in all directions Precise definitions depend on the subject area (This list is closely related to, but not identical with, the Bianchi classification of the three-dimensional real Lie algebras into nine classes . In mathematics the Bianchi classification, named for Luigi Bianchi, is a classification of the 3-dimensional real Lie algebras into 11 classes 9 of which are single In Mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable Manifolds Lie ) 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. An attractor is a set to which a Dynamical 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, renormalized to preserve volume. (Under the unrenormalized Ricci flow, the manifold collapses to a point in finite time. ) 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. (A strange and interesting fact is that all closed three-manifolds admit metrics with negative Ricci curvatures! This was proved by L. Zhiyong Gao and Shing-Tung Yau in 1986. ) 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. In Mathematics, the uniformization theorem for Surfaces says that any surface admits a Riemannian metric of constant Gaussian curvature. This topic is closely related to important topics in analysis, number theory, dynamical systems, mathematical physics, and even cosmology.

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). Felix Christian Klein ( 25 April 1849 &ndash 22 June 1925) was a German Mathematician, known for his work in Group An influential research program and manifesto was published in 1872 by Felix Klein, under the title Vergleichende Betrachtungen über neuere geometrische Forschungen Thurston's geometrization conjecture states that compact 3-manifolds can be decomposed into Submanifolds that have geometric structures In particular, the result of geometrization may be a geometry that is not isotropic. Isotropy is uniformity in all directions Precise definitions depend on the subject area 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.

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

It is possible to construct a kind of moduli space of n-dimensional Riemannian manifolds, and then the Ricci flow really does give a geometric flow (in the intuitive sense of particles flowing along flowlines) in this moduli space. In Algebraic geometry, a moduli space is a geometric space (usually a scheme or an Algebraic stack) whose points represent algebro-geometric objects of In Mathematics, specifically Differential geometry, a geometric flow is the Gradient flow associated to a functional on a Manifold which has

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

$ds^2 = \exp(2 \, p(x,y)) \, \left( dx^2 + dy^2 \right)$

(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 In the most common case the function is between domains in the Complex plane )

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. In Differential geometry, the Ricci curvature tensor, named after Gregorio Ricci-Curbastro, provides one way of measuring the degree to which the geometry determined In Differential geometry, the Laplace operator can be generalized to operate on functions defined on Surfaces or more generally on Riemannian and Pseudo-Riemannian Élie Joseph Cartan ( 9 April 1869 &ndash 6 May 1951) was an influential French Mathematician, who did fundamental Take the coframe field

$\sigma^1 = \exp (p) \, dx, \; \; \sigma^2 = \exp (p) \, dy$

so that metric tensor becomes

$\sigma^1 \otimes \sigma^1 + \sigma^2 \otimes \sigma^2 = \exp(2 p) \, \left( dx \otimes dx + dy \otimes dy \right)$

Next, given an arbitrary smooth function $h (x, y)$ , compute the exterior derivative

$d h = h_x dx + h_y dy = \exp(-p) h_x \, \sigma^1 + \exp(-p) h_y \, \sigma^2$

Take the Hodge dual

$\star d h = -\exp(-p) h_y \, \sigma^1 + \exp(-p) h_x \, \sigma^2 = -h_y \, dx + h_x \, dy$

Take another exterior derivative

$d \star d h = -h_{yy} \, dy \wedge dx + h_{xx} \, dx \wedge dy = \left( h_{xx} + h_{yy} \right) \, dx \wedge dy$

(where we used the anti-commutative property of the exterior product). In the mathematical field of Differential geometry, a metric tensor is a type of function defined on a Manifold (such as a Surface in space In Differential geometry, the exterior derivative extends the concept of the differential of a function which is a form of degree zero to Differential forms In Mathematics, the Hodge star operator or Hodge dual is a significant Linear map introduced in general by W That is,

$d \star d h = \exp(-2 p) \, \left( h_{xx} + h_{yy} \right) \, \sigma^1 \wedge \sigma^2$

Taking another Hodge dual gives

$\Delta h = \star d \star d h = \exp(-2 p) \, \left( h_{xx} + h_{yy} \right)$

which gives the desired expression for the Laplace/Beltrami operator

$\Delta = \exp(-2 \, p(x,y)) \left( D_x^2 + D_y^2 \right)$

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

$d \sigma^1 = p_y \exp(p) dy \wedge dx = -\left( p_y dx \right) \wedge \sigma^2 = -{\omega^1}_2 \wedge \sigma^2$

$d \sigma^2 = p_x \exp(p) dx \wedge dy = -\left( p_x dy \right) \wedge \sigma^1 = -{\omega^2}_1 \wedge \sigma^1$

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

${\omega^1}_2 = p_y dx - p_x dy$

Take another exterior derivative

$d {\omega^1}_2 = p_{yy} dy \wedge dx - p_{xx} dx \wedge dy = -\left( p_{xx} + p_{yy} \right) \, dx \wedge dy$

This gives the curvature two-form

${\Omega^1}_2 = -\exp(-2p) \left( p_{xx} + p_{yy} \right) \, \sigma^1 \wedge \sigma^2 = -\Delta p \, \sigma^1 \wedge \sigma^2$

from which we can read off the only linearly independent component of the Riemann tensor using

${\Omega^1}_2 = {R^1}_{212} \, \sigma^1 \wedge \sigma^2$

Namely

${R^1}_{212} = -\Delta p$

from which the only nonzero components of the Ricci tensor are

R 22 = R 11 = - Δ p

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

$R_{xx} = R_{yy} = -\left( p_{xx} + p_{yy} \right)$

But the metric tensor is also diagonal, with

g x x = g y y = exp(2 p)

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

$\frac{\partial \log p}{\partial t} = \Delta \log p$

This is manifestly analogous to the best known of all diffusion equations, the heat equation

$\frac{\partial u}{\partial t} = \Delta u$

where now $\Delta = D_x^2 + D_y^2$ is the usual Laplacian on the Euclidean plane. In the Mathematical field of Differential geometry, the Riemann curvature tensor or Riemann–Christoffel tensor is the most standard way to express In Differential geometry, the Ricci curvature tensor, named after Gregorio Ricci-Curbastro, provides one way of measuring the degree to which the geometry determined The heat equation is an important Partial differential equation which describes the distribution of Heat (or variation in temperature in a given region over time In Mathematics and Physics, the Laplace operator or Laplacian, denoted by \Delta\ or \nabla^2 and named after The reader may object that the heat equation is of course a linear partial differential equation--- where is the promised nonlinearity in the p. The word linear comes from the Latin word linearis, which means created by lines. In Mathematics, partial differential equations ( PDE) are a type of Differential equation, i d. e. defining the Ricci flow?

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. In Mathematics, a singularity is in general a point at which a given mathematical object is not defined or a point of an exceptional set where it fails to be 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 $t 0$ . In Mathematics, and more specifically in Algebraic topology and Polyhedral combinatorics, the Euler characteristic is a Topological invariant In certain cases such neckpinches will produce manifolds called Ricci solitons.

There are many related geometric flows, some of which (such as the Yamabe flow and the Calabi flow) has properties similar to the Ricci flow. In Mathematics, specifically Differential geometry, a geometric flow is the Gradient flow associated to a functional on a Manifold which has In Differential geometry, the Yamabe flow is an intrinsic Geometric flow —a process which deforms the metric of a Riemannian manifold. In Differential geometry, the Calabi flow is an intrinsic Geometric flow —a process which deforms the metric of a Riemannian manifold —in a

References

Bruce Kleiner and John Lott, Notes on Perelman's Papers (May 2006). In Mathematics, the uniformization theorem for Surfaces says that any surface admits a Riemannian metric of constant Gaussian curvature. Thurston's geometrization conjecture states that compact 3-manifolds can be decomposed into Submanifolds that have geometric structures This entry describes the solution of the Poincaré conjecture at a level intended for the general public In Differential geometry, the Ricci curvature tensor, named after Gregorio Ricci-Curbastro, provides one way of measuring the degree to which the geometry determined Calculus of variations is a field of Mathematics that deals with functionals, as opposed to ordinary Calculus which deals with functions. In Mathematics, specifically Differential geometry, a geometric flow is the Gradient flow associated to a functional on a Manifold which has Bruce Kleiner is an American Mathematician, working in Differential geometry and topology and Geometric group theory. John Richard Lott Jr (born
Cao, Huai-Dong; Xi-Ping Zhu (June 2006). Selected bibliography Huai-Dong Cao and Xi-Ping Zhu " A Complete Proof of the Poincaré and Geometrization Conjectures - application of the Hamilton-Perelman theory Award In December 2004, Zhu won the Morningside Medal of Mathematics at the Third International Congress of Chinese Mathematicians (ICCM a triennial congress hosted by "A Complete Proof of the Poincaré and Geometrization Conjectures - application of the Hamilton-Perelman theory of the Ricci flow" (PDF). Asian Journal of Mathematics 10. Erratum. Revised version (December 2006): Hamilton-Perelman's Proof of the Poincaré Conjecture and the Geometrization Conjecture
John Morgan and Gang Tian Ricci Flow and the Poincaré Conjecture (July 2006). John Willard Morgan is an American Mathematician, well-known for his contributions to Topology and Geometry. Tian Gang ( 1958 &ndash is a Chinese Mathematician and an Academician of the Chinese Academy of Sciences.
Anderson, Michael T. Geometrization of 3-manifolds via the Ricci flow, Notices AMS 51 (2004) 184--193.
John Milnor, Towards the Poincaré Conjecture and the classification of 3-manifolds, Notices AMS. John Willard Milnor (b February 20, 1931 in Orange New Jersey) is an American Mathematician known for his work in Differential 50 (2003) 1226--1233.
John Morgan, Recent progress on the Poincaré conjecture and the classification of 3-manifolds, Bull. AMS 42 (2005) 57--78.
Notes from the Clay math institute Summer School Program 2005 on Ricci flow.
Richard Hamilton, Three-manifolds with positive Ricci curvature, J. Richard Streit Hamilton (born 1943 is professor of Mathematics at Columbia University. Diff. Geom 17 (1982), 255-306.
Collected Papers on Ricci Flow ISBN 1-57146-110-8.
Bakas, I. . The algebraic structure of geometric flows in two dimensions. arXiv eprint server. Retrieved on July 28, 2005. Events 1540 - Thomas Cromwell is executed at the order of Henry VIII of England on charges of Treason. Year 2005 ( MMV) was a Common year starting on Saturday (link displays full calendar of the Gregorian calendar.

Chow, Bennet and Knopf, Dan (2004). The Ricci Flow: an introduction. American Mathematical Society. ISBN 0-8218-3515-7. .
Peter Topping (2006). Lectures on the Ricci flow. C. U. P. . ISBN 0-521-68947-3.
Tao, T. , “Ricci flow”, in Gowers, Princeton companion to mathematics, ISBN 978-0691118802, <http://terrytao.files.wordpress.com/2008/03/ricci.pdf>
Weeks, Jeffrey R. (1985). Jeffrey Weeks may refer to Jeffrey Weeks (sociologist Jeffrey Weeks (mathematician The Shape of Space: how to visualize surfaces and three-dimensional manifolds. New York: Marcel Dekker. ISBN 0-8247-7437-X. . A popular book which explains the background for the Thurston classification programme.

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Contents

Mathematical definition

Examples

Relationship to uniformization and geometrization

Relation to diffusion

Recent developments

See also

Applications

General context

References