for a Riemannian metric, 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).
Collins, G. P. "The Shapes of Space." Sci. Amer.291, 94-103, July 2004.Hamilton, R. S. "Three
Manifolds with Positive Ricci Curvature." J. Diff. Geom.17, 255-306,
1982.Hamilton, R. S. "Four Manifolds with Positive Curvature
Operator." J. Diff. Geom.24, 153-179, 1986.Kleiner,
B. and Lott, J. "Notes and Commentary on Perelman's Ricci Flow Papers."
http://www.math.lsa.umich.edu/research/ricciflow/perelman.html.Perelman,
G. "The Entropy Formula for the Ricci Flow and Its Geometric Application"
11 Nov 2002. http://arxiv.org/abs/math.DG/0211159.Robinson,
S. "Russian Reports He Has Solved a Celebrated Math Problem." The New
York Times, p. D3, April 15, 2003.Rubinstein, J. H. and
Sinclair, R. "Visualizing Ricci Flow of Manifolds of Revolution." Exp.
Math.14, 285-298, 2005.
, 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).
