4

1. THE RICCI FLOW OF SPECIAL GEOMETRIES

AT3

is closed, irreducible, and homotopic to a hyperbolic 3-manifold, then

J\f3

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 G

Mn

there exist

neighborhoods Ux C

Mn

of x and Uy C

Mn

of y and a g-isometry j

x y

:

Ux — Uy such that j

x y

(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,i/G

A4n

there exists a p-isometry

Jxy •

M71

~*

M.n

such that j

x y

{x) = y. These concepts are equivalent when

M.n

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,

#), 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

(A1n,

Q, ?*), where

Ain

is a simply-connected smooth manifold and Q is a group of diffeo-

morphisms that acts smoothly and transitively on

M71

such that for each

x G A4n, the isotropy group (point stabilizer)

is isomorphic to Q*. We say a model geometry

(A4n,G,G*)

is a maximal

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

phism group 2)

(Mn)

that have compact isotropy groups. (An example of

particular relevance occurs when

Mn

= Q 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

(Mnyg).

PROOF.

One may obtain a complete homogeneous ^-invariant Riemann-

ian metric on Mn as follows. Choose any x G Mn. ligx is any scalar product