6 1. THE RICCI FLOW OF SPECIAL GEOMETRIES
COROLLARY 1.11.
( 1) Given x, y E Mn and a linear isometry
f: (TxMn, g (x))---+ (TyMn, g (y)),
there exists at most one isometry 'Y E Isom (Mn, g) with 'Y ( x) = yand 'Y* = f.
(2) If Mn is compact, then Isom(Mn,g) is compact.When (Mn, g) is homogeneous, all isotropy groups look the same.
LEMMA 1.12. If Isom (Mn, g) acts transitively on Mn, then all isotropy
groups Ix (Mn,g) are isomorphic.
PROOF. Given any x, y E Mn, there exists an isometry 'Y : Mn ---+ Mn
such that 'Y (x) = y. Define
'Y#: Ix (Mn,g)---+ Iy (Mn,g)
by
'Y# ((3) ~ 'Y 0 (3 0 'Y-^1.Clearly, 'Y# is a group isomorphism with inverse ('Y-^1 ) #. D
Thus one may regard a complete homogeneous space (Mn, g) as a model
geometry (Mn,Isom(Mn,g) ,Ix (Mn,g)).REMARK 1.13. There is an ambiguity in the equivalence of model geome-
tries and homogeneous models that we now address. The variety of possible
Riemannian metrics g compatible with a given model geometry (Mn, g, g*)
depends strongly on the size of the isotropy group(}*. If the isotropy group
g* is large enough, for example if g* ~ 0 (n), then one may show easily that
a unique (}*-invariant metric g, hence a unique homogenous model (Mn, g)
is determined up to a scalar multiple. For smaller isotropy groups g*, there
may be more choices for g, and these may endow Mn with somewhat dif-
ferent geometric properties. On the other hand, there may be closed proper
subgroups of Isom(Mn,g) acting transitively on (Mn,g), so that a homo-
geneous model (Mn, g) may be regarded as a model geometry (Mn, g, g*)
under different groups (Mn,g',(}~). For the purpose of giving a standard
description of the geometric structures that occur in the Geometrization
Conjecture, this ambiguity is addressed by choosing the isotropy groups to
be as large as possible, in other words by considering only maximal model
geometries. In fact, Thurston proved that there are exactly eight maximal
model geometries that have compact 3-manifold representatives.- Classifying three-dimensional maximal model geometries
In dimension three, a surprisingly rich collection of model geometries
is obtained by taking M^3 = (}^3 to be a Lie group acting transitively on
itself by multiplication. Since we are interested only in closed manifolds
modeled on (}^3 , we can make a further restriction. One says a Lie group