The question of which Riemannian manifolds admit simple closed
geodesics is still
a mystery. It is not known whether all closed Riemannian manifolds contain
simple
closed geodesics. For closed manifolds with nontrivial fundamental group,
a simple
closed geodesic can always be found by taking the shortest homotopically
nontrivial
closed geodesic. When the manifold is closed but simply connected, the
question is
open for dimensions three and above. In dimension two, it is known by the
theorem
of Lusternik and Schnirelmann [6] (see also [3]
and
[4]) that the 2-sphere equipped
with any smooth Riemannian metric contains at least three distinct simple
closed
geodesics.
Non-compact manifolds do not necessarily contain closed geodesics, Euclidean
space being an obvious example. Even if the manifold is not simply connected,
it may
not contain any simple closed geodesics, as with the hyperbolic thrice-punctured
sphere. However, among the orientable, finite area, complete hyperbolic
2-manifolds,
the thrice-punctured sphere is the only example that contains no simple
closed
geodesic. In this paper, we shall determine which orientable hyperbolic
3-manifolds
do and do not contain simple closed geodesics. We shall prove that the
Fuchsian
group corresponding to the thrice-punctured sphere generates the only example
of a
complete non-elementary orientable hyperbolic 3-manifold that does not
contain a
simple closed geodesic. We do not assume that the manifold is geometrically
finite, or
even that it has finitely generated fundamental group. The simple closed
geodesic
which we produce arises from an interesting class of elements of the fundamental
group. It is the shortest closed geodesic corresponding to a screw motion
induced by
the action of the fundamental group on hyperbolic 3-space.
In proving our result, we use geometric methods to obtain results about
isometries
of hyperbolic 3-space. These results apply to elliptic, as well as to parabolic
and
loxodromic isometries, and we state them in full generality.
A related question is whether a hyperbolic 3-manifold always contains
a non-simple
closed geodesic. In [2], Alan Reid and Ted Chinburg
utilized arithmetic
hyperbolic 3-manifold theory to construct examples of closed hyperbolic
3-manifolds
in which every closed geodesic is simple.
We shall be working with orientable 3-manifolds, so all of the isometries
discussed
will be orientation preserving.