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Ricci flow

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. It plays an important role in [Grigori Perelman's] [solution of the PoincarÃ© conjecture] ; in this context is also called the Ricciâ€“Hamilton flow .

Mathematical definition

Given a Riemannian manifold with [metric tensor] g_{ij}, we can compute the
[Ricci tensor] R_{ij}, 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

:\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_\mathrm{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.

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