1547671870-The_Ricci_Flow__Chow

4 1. THE RICCI FLOW OF SPECIAL GEOMETRIES

N^3 is closed, irreducible, and homotopic to a hyperbolic 3-manifold, then
N^3 is homeomorphic to a hyperbolic 3-manifold.

2. Model geometries

A geometric structure on an n-dimensional manifold Mn may be

regarded as a complete locally homogeneous Riemannian metric g. A metric

is locally homogeneous if it looks the same at every point. More precisely,

one says that g is locally homogeneous if for all x, y E Mn there exist

neighborhoods Ux ~ Mn of x and Uy ~ Mn of y and a g-isometry 'Yxy :

Ux ----+ Uy such that 'Yxy (x) = y. In this section, we shall develop a more

abstract way to think about geometric structures.

One says that a Riemannian metric g on Mn is homogeneous (to

wit, globally homogeneous) if for every x, y E Mn there exists a g-isometry

'Yxy : Mn----+ Mn such that 'Yxy (x) = y. These concepts are equivalent when

Mn is simply connected.

PROPOSITION 1. 7 (Singer [120]). Any locally homogeneous metric g on

a simply-connected manifold is globally homogeneous.

We also have the following well-known fact.

PROPOSITION 1.8. If (Mn, g) is homogeneous, then it is complete.

Thus if (Mn, g) is locally homogeneous, we say that its geometry is given

by the homogeneous model (Mn, g), where Mn is the universal cover of

Mn and g is the complete metric obtained by lifting the metric g. Thus to

study geometric structures, it suffices to study homogeneous models.

A model geometry in the sense of Klein is a tuple (Mn, g, 9*), where

Mn is a simply-connected smooth manifold and g is a group of diffeo-

morphisms that acts smoothly and transitively on Mn such that for each

x E Mn, the isotropy group (point stabilizer)

9x ~ { 'Y E g : "( ( x) = x}

is isomorphic to g*. We say a model geometry (Mn, 9, 9*) is a maximal

model geometry if g is maximal among all subgroups of the diffeomor-

phism group ::D (Mn) that have compact isotropy groups. (An example of

particular relevance occurs when Mn = g is a 3-dimensional unimodular

Lie group. These maximal models were classified by Milnor in [98] and are

briefly reviewed in Section 4 of this chapter.)

The concepts of model geometry and homogeneous model are essentially

equivalent (up to a subtle ambiguity that will be addressed in Remark 1.13).

One direction is easy to establish.

LEMMA 1.9. Every model geometry (Mn,g,g*) may be regarded as a

complete homogeneous space (Mn, g).

PROOF. One may obtain a complete homogeneous 9 -invariant Riemann-

ian metric on Mn as follows. Choose any x E Mn. If §x is any scalar product