Changes

The effect of the Ricci flow has much in common mathematically with the diffusion of heat in a surface, a concept called [[entropy]]. For example, look at the heat diffusion equation <math>u_t = \Delta u \ </math>.  This equation is precisely an expression for the change in the metric of a manifold over time in Ricci flow if we take <math>u = \ln g_{jj} \ </math>, where j is any number less than the dimension of the manifold, no summation is implied on that index, and g is a metric symmetric on the main diagonal (any relevant manifold can have a coordinate system which creates such a metric).

The effect of the Ricci flow has much in common mathematically with the diffusion of heat in a surface, a concept called [[entropy]]. For example, look at the heat diffusion equation <math>u_t = \Delta u \ </math>.  This equation is precisely an expression for the change in the metric of a manifold over time in Ricci flow if we take <math>u = \ln g_{jj} \ </math>, where j is any number less than the dimension of the manifold, no summation is implied on that index, and g is a metric symmetric on the main diagonal (any relevant manifold can have a coordinate system which creates such a metric).

Noticing this (and other similarities between the effect of Ricci flow on a manifolds metric and the effect of time on heat dispersion) Perelman formulated the concept of "Perelman entropy."   

Noticing this (and other similarities between the effect of Ricci flow on a manifolds metric and the effect of time on heat dispersion), Perelman formulated the concept of "Perelman entropy."   

[[Category:Mathematics]]

[[Category:Mathematics]]