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The Princeton Companion to Mathematics by Imre Leader, June Barrow-Green, Timothy Gowers

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III.78 Ricci Flow

  Terence Tao

Ricci flow is a technique that allows one to take an arbitrary RIEMANNIAN MANIFOLD [I.3 §6.10] and smooth out the geometry of that manifold to make it look more symmetric. It has proven to be a very useful tool in understanding the topology of such manifolds.

Ricci flow can be defined for Riemannian manifolds of any dimension, but for the sake of exposition we restrict ourselves here to two-dimensional manifolds (i.e., surfaces) as they are easy to visualize. From our everyday experience with three-dimensional space Image3, we are familiar with many surfaces, such as spheres, cylinders, planes, tori (the shape of the ...

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