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, where 
is the Ricci curvature
tensor. Hamilton (1982) showed that there is a unique solution to this equation
for an arbitrary smooth metric on a closed manifold
over a sufficiently short time. Hamilton (1982, 1986) also showed that Ricci flow
preserves positivity of the Ricci curvature
tensor in three dimensions and the curvature
operator in all dimensions (Perelman 2002).
