The proof followed on from the program of Richard Hamilton to use the Ricci flow to attack the problem.

Perelman introduced a modification of the standard Ricci flow, called Ricci flow with surgery to systematically excise singular regions as they develop, in a controlled way.

On July 1, 2010, he turned down the prize saying that he believes his contribution in proving the Poincaré conjecture was no greater than that of Hamilton's ( who first suggested using the Ricci flow for the solution ).

Several stages of the Ricci flow on a two-dimensional manifold.

Hamilton's program was started in his 1982 paper in which he introduced the Ricci flow on a manifold and showed how to use it to prove some special cases of the Poincaré conjecture.

In these papers he sketched a proof of the Poincaré conjecture and a more general conjecture, Thurston's geometrization conjecture, completing the Ricci flow program outlined earlier by Richard Hamilton.

The metric is improved using the Ricci flow equations ;

Ricci flow expands the negative curvature part of the manifold and contracts the positive curvature part.

In some cases Hamilton was able to show that this works ; for example, if the manifold has positive Ricci curvature everywhere he showed that the manifold becomes extinct in finite time under Ricci flow without any other singularities.

However in general the Ricci flow equations lead to singularities of the metric after a finite time.

Perelman showed how to continue past these singularities: very roughly, he cuts the manifold along the singularities, splitting the manifold into several pieces, and then continues with the Ricci flow on each of these pieces.

This procedure is known as Ricci flow with surgery.

A special case of Perelman's theorems about Ricci flow with surgery is given as follows.

Perelman's proof uses a modified version of a Ricci flow program developed by Richard Hamilton.

Perelman proved the conjecture by deforming the manifold using something called the Ricci flow ( which behaves similarly to the heat equation that describes the diffusion of heat through an object ).

The Ricci flow usually deforms the manifold towards a rounder shape, except for some cases where it stretches the manifold apart from itself towards what are known as singularities.

Major steps in the proof involve showing how manifolds behave when they are deformed by the Ricci flow, examining what sort of singularities develop, determining whether this surgery process can be completed and establishing that the surgery need not be repeated infinitely many times.

The first step is to deform the manifold using the Ricci flow.

The Ricci flow was defined by Richard Hamilton as a way to deform manifolds.

The formula for the Ricci flow is an imitation of the heat equation which describes the way heat flows in a solid.

Like the heat flow, Ricci flow tends towards uniform behavior.

Matteo Ricci was among the very earliest to report on the thoughts of Confucius, and father Prospero Intorcetta wrote about the life and works of Confucius in Latin in 1687.

The Latinized name " Confucius " is derived from " Kong Fuzi ", which was first coined by 16th-century Jesuit missionaries to China, most probably by Matteo Ricci.

An early European account of Taoism was provided by the Jesuits Matteo Ricci and Nicolas Trigault in their De Christiana expeditione apud Sinas ( 1615 ).

While the transcription of the Chinese words used by Ricci was not very consistent, he systematically used Latin p and t for unaspirated Chinese sounds that Pinyin renders as b and d. Accordingly, Ricci called the adherents of Laozi, Tausu (, Pinyin: Daoshi ), which was rendered as Tausa in an early English translation published by Samuel Purchas ( 1625 ).

Instead of trying to approach Christianity through the traditions of the local religion and creating a nativised church as latter fellow Jesuit Matteo Ricci did in China, he was eager for change.

The Matteo Ricci College was founded in 1973 and named after Italian Jesuit missionary, Matteo Ricci.

Tensor calculus was developed around 1890 by Gregorio Ricci-Curbastro under the title absolute differential calculus, and originally presented by Ricci in 1892.

It was made accessible to many mathematicians by the publication of Ricci and Tullio Levi-Civita's 1900 classic text Méthodes de calcul différentiel absolu et leurs applications ( Methods of absolute differential calculus and their applications ).

A derivative, Daoshi (, " Daoist priest "), was used already by the Jesuits Matteo Ricci and Nicolas Trigault in their De Christiana expeditione apud Sinas, rendered as Tausu in the original Latin edition ( 1615 ), and Tausa in an early English translation published by Samuel Purchas ( 1625 ).

* In 2006 a two-episode TV movie was broadcast by Italian state television RAI, dedicated to the magistrate, starring Massimo Dapporto as Falcone and Elena Sofia Ricci as his wife Francesca Morvillo.

By c. 1600, the Jesuits stationed in China, led by Matteo Ricci, were pretty sure that it was, but others were not convinced yet.

Ultimately, the Salt Lake City police signaled that their prime person of interest was Richard Ricci, being held in custody for unrelated reasons.

Ricci, a handyman hired by the Smarts, was on parole for a 1983 attempted murder of police officer Mike Hill.

In 2005 the band was the subject of a documentary called Fearless Freaks, featuring appearances by other artists and celebrities such as Gibby Haynes, The White Stripes, Beck, Christina Ricci, Liz Phair, Juliette Lewis, Steve Burns, Starlight Mints, and Adam Goldberg.

* Scenes from the motion picture After. Life, starring Christina Ricci, Liam Neeson and Justin Long, was filmed on Atlantic Ave. downtown district and on Merrick Rd.

Apart from having to lose weight, Ricci ate only food items without any nutritional value to achieve her unhealthy appearance.

There is an ( n − 2 )- dimensional family of such 2-planes, and so only in dimensions 2 and 3 does the Ricci tensor determine the full curvature tensor.

This happens if and only if the Ricci tensor Ric is a constant multiple of the metric tensor g.

# Myers ' theorem states that if the Ricci curvature is bounded from below on a complete Riemannian manifold by, then the manifold has diameter, with equality only if the manifold is isometric to a sphere of a constant curvature k. By a covering-space argument, it follows that any compact manifold of positive Ricci curvature must have finite fundamental group.

In particular, given a point p in a Riemannian manifold, it is always possible to find metrics conformal to the given metric g for which the Ricci tensor vanishes at p. Note, however, that this is only pointwise assertion ; it is usually impossible to make the Ricci curvature vanish identically on the entire manifold by a conformal rescaling.

If n 3, the trace-free Ricci tensor vanishes identically if and only if

In this more general situation, the Ricci tensor is symmetric if and only if there exist locally a parallel volume form for the connection.

Unlike the Ricci tensor and sectional curvature, however, global results involving only the scalar curvature are extremely subtle and difficult.

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.

) 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.

from which the only nonzero components of the Ricci tensor are

So if 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.

Ricci mentioned this man's name as Ngai, who has since been identified by the French sinologist Paul Pelliot as a Jew named Ai T ' ien, who explained that the community he belonged to was monotheistic, or believing in only one God.

One need only to look at his face, his uncertain gait, his hesitant or fearful attitudes to understand that Ricci is already a victim, a diminished man who has lost his confidence.

* One of the most highly rated of Martin's programs was a Christmas episode featuring only his and Frank Sinatra's family members: Martin's wife Jeanne with children Craig, Claudia, Gail, Deana, Dean Paul, Ricci and Gina along with Sinatra's three children, Tina, Nancy and Frank Jr.

The Ricci tensor depends only on the metric tensor, so the Einstein tensor can be defined directly with just the metric tensor.

It is a mathematical fact that the Einstein tensor vanishes if and only if the Ricci tensor vanishes.

A third equivalent condition follows from the Ricci decomposition of the Riemann curvature tensor as a sum of the Weyl curvature tensor plus terms built out of the Ricci tensor: the Weyl and Riemann tensors agree,, in some region if and only if it is a vacuum region.

the only nonvanishing component of the Ricci spinor is

From the form of Ricci spinor given in the preceding section, it is immediately apparent that a pp-wave spacetime ( written in the Brinkmann chart ) is a vacuum solution if and only if is a harmonic function ( with respect to the spatial coordinates ).