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.

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.

For two dimensional manifolds, the above formula shows that if f is a harmonic function, then the conformal scaling g e < sup > 2ƒ </ sup > g does not change the Ricci curvature.

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.

) 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 negatively curved two-manifold into a two-dimensional multi-holed torus which is locally isometric to the hyperbolic plane.

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.

The sign of the Ricci tensor does not change, because the two sign conventions concern the sign of the Riemann tensor.

With four or more dimensions, Ricci curvature does not describe the curvature tensor completely.

There is now a memorial plaque in Zhaoqing to commemorate Ricci's six-year stay there as well as a building set up as a " Ricci Memorial Centre " although the building itself does not date back to the time of the priest as it was built in the 1860s.

Since giving the Einstein tensor does not fully determine the Riemann tensor, but leaves the Weyl tensor unspecified ( see the Ricci decomposition ), the Einstein equation may be considered a kind of compatibility condition: the spacetime geometry must be consistent with the amount and motion of any matter or nongravitational fields, in the sense that the immediate presence " here and now " of nongravitational energy-momentum causes a proportional amount of Ricci curvature " here and now ".

For the duration of World War II, Dior, as an employee of Lelong — who labored to preserve the French fashion industry during wartime for economic and artistic reasons — designed dresses for the wives of Nazi officers and French collaborators, as did other fashion houses that remained in business during the war, including Jean Patou, Jeanne Lanvin, and Nina Ricci.

Affines also preserve the Riemann, Ricci and Weyl tensors, i. e.

Drawing further upon the analogy with geometric Newtonian gravity, it is natural to assume that the field equation for gravity relates this tensor and the Ricci tensor, which describes a particular class of tidal effects: the change in volume for a small cloud of test particles that are initially at rest, and then fall freely.

The volume of a metric ball of radius r in a complete n-dimensional Riemannian manifold with positive Ricci curvature has volume at most that of the volume of a ball of the same radius r in Euclidean space.

In differential geometry, the Ricci curvature tensor, named after Gregorio Ricci-Curbastro, represents the amount by which the volume element of a geodesic ball in a curved Riemannian manifold deviates from that of the standard ball in Euclidean space.

Similarly, if the Ricci curvature is negative in the direction of a given vector ξ, such a conical region in the manifold will instead have larger volume than it would in Euclidean space.

# The Bishop – Gromov inequality states that if a complete m-dimensional Riemannian manifold has non-negative Ricci curvature, then the volume of a ball is smaller or equal to the volume of a ball of the same radius in Euclidean m-space.

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

with R < sub > AB </ sub > the five-dimensional Ricci curvature, may be re-expressed so that in four dimensions, these solutions satisfy Einstein's equations

He wanted to cut the manifold at the singularities and paste in caps, and then run the Ricci flow again, so he needed to understand the singularities and show that certain kinds of singularities do not occur.

Abelian surfaces have a Ricci flat metric with holonomy strictly smaller than SU ( 2 ) ( in fact trivial ) so are not Calabi – Yau manifolds according to such definitions.

Roughly speaking, the Ricci flow contracts positive curvature regions and expands negative curvature regions, so it should kill off the pieces of the manifold with the " positive curvature " geometries S < sup > 3 </ sup > and S < sup > 2 </ sup > × R, while what is left at large times should have a thick-thin decomposition into a " thick " piece with hyperbolic geometry and a " thin " graph manifold.

If the Ricci tensor vanishes, then the canonical bundle is flat, so the structure group can be locally reduced to a subgroup of the special linear group SL ( n, C ).

However, Kähler manifolds already possess holonomy in U ( n ), and so the ( restricted ) holonomy of a Ricci flat Kähler manifold is contained in SU ( n ).

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.

Jerry Falk ( Biggs ) is an aspiring writer living in New York City who falls in love at first sight with Amanda ( Ricci ) and begins having an affair and eventually tells his girlfriend about it so that she will dump him because Falk cannot end relationships.

The Jesuits at the time were interested in expanding their missionary activity from Macau into Mainland China, so other visits by Jesuits soon followed, and by 1583, after several false starts, Michele Ruggieri and another, recently arrived Jesuit, named Matteo Ricci managed to establish residence in the city-the first Jesuit mission house in China outside Macau.

for a constant ρ ∈ R. Further, let M < sub > k </ sub >< sup > n </ sup > be the n-dimensional simply connected Riemannian space form of constant sectional curvature k = ( n-1 ) ρ ( i. e. constant Ricci curvature ρ ), so M < sub > k </ sub >< sup > n </ sup > is an n-sphere if k > 0, it is an n-dimensional Euclidean space if k = 0, and it is an n-dimensional hyperbolic space if k < 0.

The Ricci curvature, or trace component of the Riemann tensor contains precisely the information about how volumes change in the presence of tidal forces, so the Weyl tensor is the traceless component of the Riemann tensor.

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

The Einstein field equations — which determine the geometry of spacetime in the presence of matter — contain the Ricci tensor, and so calculating the Christoffel symbols is essential.

Ricci was not able to come to the recording studio, so she recorded all of her lines over the phone.

If we adopt Cartesian coordinates with line element with corresponding wave operator on the flat background, or Minkowski spacetime, so that the line element of the curved spacetime is, then the Ricci scalar of this curved spacetime is just

As Sogang began life as a full university, there was an increase in student numbers and the Library, which had moved to Ricci Hall, became extremely congested with students, so there was an urgent need for an independent library building.