1547845439-The_Ricci_Flow_-_Techniques_and_Applications_-_Part_I__Chow_

510 C. GLOSSARY

fields. Formally it is defined by

Rm (X, Y) Z ~ Y'x\i'yZ - \7y\7xZ - Y'[x,Y]Z.

Rosenau solution. An explicit rotationally symmetric ancient solution

on the 2-sphere with positive curvature. Its backward limit as t ---+ -oo

(without rescaling) is the cigar soliton.

scalar curvature. The trace of the Ricci tensor: R = l.:~ 1 Re (ei, ei) =

gij~j·

sectional curvature. The number

K (P) = (Rm (e1, e2) e2, e1)

associated to a 2-plane in a tangent space PC TxM, where {e1, e2} is an

orthonormal basis of P.

self-similar solution. (For the Ricci fl.ow, see Ricci soliton.)

Shi's local derivative estimate. A local estimate for the covariant

derivatives of the Riemann curvature tensor.

short-time existence. The existence, when it holds, of a solution to

the initial-value problem for the Ricci fl.ow on some nontrivial time interval.

For example, for a smooth initial metric on a closed manifold.

shrinking gradient Ricci soliton ( a.k.a. shrinker). A gradient Ricci

soliton which is evolving by the pull-back by diffeomorphisms and scalings

less than 1.

singular solution. A solution on a maximal time interval. If the

maximal time interval [O, T) is finite, then

sup [Rm[ < oo.

Mnx[O,T)

singularity. For example, if T is the singular (i.e., maximal) time, we

say that the solution forms a singularity at time T.
singularity model. The limit of dilations of a singular solution. For
finite time singular solutions, singularity models are ancient solutions. For
infinite time singular solutions, singularity models are immortal solutions.
singularity time. For a solution, the time T E (0, oo], where [O, T) is
the maximal time interval of existence.
Sobolev inequality. A class of inequalities where the Lg-norms of
functions are bounded by the .LP-norms of their derivatives, where p and q
are related by the dimension and number of derivatives.
space form. A complete Riemannian manifold with constant sectional
curvature.

space-time. The manifold Mn x I of a solution (Mn, g ( t)) defined on

the time interval I.
space-time connection. Any of a class of natural connections on the
tangent bundle of space-time
spherical space form. A complete Riemannian manifold with positive
constant sectional curvature. Such a manifold is, after scaling, isometric to
a quotient of the unit sphere by a group of linear isometries.