C. GLOSSARY 509Poincare conjecture. The conjecture that any simply-connected closed
3-manifold is diffeomorphic to the 3-sphere. (In dimension 3, the topological
and differentiable categories are the same.) Hamilton's program and Perel-
man's work aim to complete a proof of the Poincare conjecture using Ricci
fl.ow.
positive (nonnegative) curvature operator. The eigenvalues of the
curvature operator are positive (nonnegative).
potentially infinite dimensions. A device which Perelman combined
with the space-time approach to the Ricci flow to embed solutions of the
Ricci flow into a potentially Ricci fl.at manifold with potentially infinite
dimension.
preconvergent sequence. A sequence for which a subsequence con-
verges.
quasi-Einstein metric. The mathematical physics jargon for a non-
Einstein gradient Ricci soliton.
Rademacher's Theorem. The result that a locally Lipschitz function
is differentiable almost everywhere.
reaction-diffusion equation. A heat-type equation consisting of the
heat equation plus a nonlinear term which is zeroth order in the solution.
reduced distance. The distance-like function for solutions of the back-
ward Ricci flow defined bye (q, T) = 2.J:rL (q, T). (See L-distance.) Partly
motivated by consideration of the heat kernel and the Li-Yau distance func-
tion for positive solutions of the heat equation.
reduced volume. For a solution to the backward Ricci fl.ow, the time-
dependent invariant
V (T) = r (47rT)-nf^2 e-£(q,T)dμg(T) (q).
}Mn
Ricci fl.ow. The equation for metrics is gt9ij = -2Rij· This equation
was discovered and developed by Richard Hamilton and is the subject of
this book.
Ricci soliton. (See also gradient Ricci soliton.) A self-similar solu-
tion of the Ricci fl.ow. That is, the solution evolves by scaling plus the Lie
derivative of the metric with respect to some vector field.
Ricci tensor. The trace of the Riemann curvature operator:
n
Re (X, Y) =trace (X f-+ Rm (X, Y) Z) = L (Rm (ei, X) Y, ei).
i=l
Ricci-DeTurck fl.ow. A modification of the Ricci flow which is a
strictly parabolic system. This equation is essentially equivalent to the Ricci
fl.ow via the pull-back by diffeomorphisms and is used to prove short-time
existence for solutions on closed manifolds.
Riemann curvature operator. (See curvature operator.)
Riemann curvature tensor. The curvature (3, 1)-tensor obtained by
anti-commuting in a tensorial way the covariant derivatives acting on vector