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
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
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 ):
and its finite quotients – lens spaces, dodecahedron spaces –
classification was completed by Hopf.
(2) Flat: T
and its finite quotients – these are completely classified.