Ricci flow
In differential geometry, the Ricci flow (/ˈriːtʃi/) is an intrinsic geometric flow. It is a process that deforms the metric of a Riemannian manifold in a way formally analogous to the diffusion of heat, smoothing out irregularities in the metric.
The Ricci flow, named after Gregorio Ricci-Curbastro, was first introduced by Richard Hamilton in 1981 and is also referred to as the Ricci–Hamilton flow. It is the primary tool used in Grigori Perelman's solution of the Poincaré conjecture, as well as in the proof of the differentiable sphere theorem by Simon Brendle and Richard Schoen.
Mathematical definition
Given a Riemannian manifold with metric tensor 
, we can compute the Ricci tensor 
, 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 geometric evolution equation
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Top Of The World
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[Chorus]
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