Lecture 1
In this course, we focus on compact and orientable 3-manifolds. We ask the
following fundamental questions:
What do all 3-manifolds look like? Can we classify them?
Consider the case of closed orientable surfaces. They are characterized by the
genus. The case g = 0 is the 2-sphere. One can equip it with the round metric. The
Figure 1. Surfaces with genus 0, 1 and 2
case g = 1 corresponds to the two dimensional torus

=
R2/Γ, where Γ is a lattice
subgroup of R2. It admits a naturally induced flat metric from R2. For g 2 we
can equip the surface Σg of genus g with the hyperbolic metric induced from the
Poincar´ e disk model of H2; that is to say there is a discrete, torsion-free subgroup
Γg of the isometries of
H2
with Σg

=
H2/Γg.
Geometric manifolds
Definition (Homogeneous metric). Let (M, g) be a Riemannian manifold.
The metric g is called homogeneous if the action Isom(M)×M M is transitive.
Definition (Locally homogeneous metric). A Riemannian metric g on a man-
ifold M is called locally homogeneous if its lifted metric ˜ g on the universal cover
˜
M is homogeneous.
The round metric of the 2-sphere (i.e. g = 0) is homogeneous, but the hyper-
bolic metric of the genus-2 Riemann surface is not. However, the latter is locally
homogeneous.
Definition (Geometric manifolds). A manifold is called geometric if it ad-
mits a finite volume complete locally homogeneous Riemannian metric.
Here is the list of Geometric 3-manifolds by type (see [19]):
(1) Round:
S3
and its finite quotients lens spaces, dodecahedron spaces
classification was completed by Hopf.
(2) Flat: T
3
and its finite quotients these are completely classified.
3
http://dx.doi.org/10.1090/ulect/053/01
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