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
