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Ricci flow expands the negative curvature part of the manifold and contracts the positive curvature part.
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Ricci and flow
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 ;
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.
Ricci and expands
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.
Ricci and negative
# The isometry group of a compact Riemannian manifold with negative Ricci curvature is discrete.
# Any smooth manifold of dimension n ≥ 3 admits a Riemannian metric with negative Ricci curvature.
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.
Briefly, positive Ricci curvature of a Riemannian manifold has strong topological consequences, while ( for dimension at least 3 ), negative Ricci curvature has no topological implications.
Ricci curvature is now known to have no topological implications ; has shown that any manifold of dimension greater than two admits a Riemannian metric of negative Ricci curvature.
( For surfaces, negative Ricci curvature implies negative sectional curvature ; but the point
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.
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.
) 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!
Authors including Einstein have used a different sign in their definition for the Ricci tensor which results in the sign of the constant on the right side being negative
The special case of 4 dimensions in physics ( 3 space, 1 time ) gives, the trace of the Einstein tensor, as the negative of, the Ricci tensor's trace.
In the case of a Lorentzian manifold,, the Einstein tensor has, by design, a trace which is just the negative of the Ricci scalar, and one may check that the traceless part of the Einstein tensor agrees with the traceless part of the Ricci tensor.
Quaternion-Kähler manifolds divide naturally into those with positive and negative Ricci curvature.
Ricci and curvature
The Ricci tensor itself is related to the more general Riemann curvature tensor as
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
where g is the metric and R its Ricci curvature,
) This easily implies the Poincaré conjecture in the case of positive Ricci curvature.
Two more generalizations of curvature are the scalar curvature and Ricci curvature.
The difference in area of a sector of the disc is measured by the Ricci curvature.
Each of the scalar curvature and Ricci curvature are defined in analogous ways in three and higher dimensions.
This is the classical method used by Ricci and Levi-Civita to obtain an expression for the Riemann curvature tensor.
It is also exactly half the scalar curvature of the 2-manifold, while the Ricci curvature tensor of the surface is simply given by
* M has a Kähler metric with vanishing Ricci curvature.
This follows from Yau's proof of the Calabi conjecture, which implies that a compact Kähler manifold with a vanishing first real Chern class has a Kähler metric in the same class with vanishing Ricci curvature.
If a compact Riemannian manifold has positive Ricci curvature then its fundamental group is finite.
If a complete n-dimensional Riemannian manifold has nonnegative Ricci curvature and a straight line ( i. e. a geodesic which minimizes distance on each interval ) then it is isometric to a direct product of the real line and a complete ( n-1 )- dimensional Riemannian manifold which has nonnegative Ricci curvature.
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.
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