Ricci Flow - Mathematical Definition

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

The normalized Ricci flow makes sense for compact manifolds and is given by the equation

where is the average (mean) of the scalar curvature (which is obtained from the Ricci tensor by taking the trace) and is the dimension of the manifold. This normalized equation preserves the volume of the metric.

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. (This is similar to the way in which the heat equation can be run forwards in time, but not usually backwards in time.)

Informally, the Ricci flow tends to expand negatively curved regions of the manifold, and contract positively curved regions.