Let M be a three dimensional Riemannian Manifold that is compact. .

[Reverie]

Let M be a three dimensional Riemannian Manifold that is compact and does not have boundary. I believe manifolds that are compact and without boundary are called closed. So, my manifold M is closed.

I'm interested in knowing the answers to the following questions.

Under what conditions is there a unique closed geodesic in each free homotopy class of loops?
a.)If the universal cover is isometric to the manifolds occurring in Thurston's geometrization theorem and the homotopy class is non-trivial, is there exactly one closed geodesic in each free homotopy class of loops? This is true for hyperbolic manifolds. Is it true for other manifolds ocurring in the geometrization theorem?
b.)When a homotopy class does not have a unique closed geodesic, can anything be said about the number of closed geodesics in that free homotopy class? For example, are there infinitely many geodesics? Are the lengths of the geodesics related?

I believe the following facts are true.
1.)There is a closed geodesic in each nontrivial free homotopy class of loops.
2.)For hyperbolic manifolds, a unique geodesic exists in each free homotopy class of loops.

I would like to know any easily stated theorem describing the structure of closed geodesics in three manifolds. I would also like to know easily stated conjectures, whose answers are unknown.

Thanks
-Reverie

 

[jvicens]

Dear Reverie,

Thank you for your interesting questions regarding closed geodesics in three-dimensional Riemannian manifolds. I will address your questions in order and provide some relevant theorems and conjectures that are currently being researched in this area.

1. Under what conditions is there a unique closed geodesic in each free homotopy class of loops?

As you mentioned, for hyperbolic manifolds, there is a unique closed geodesic in each free homotopy class of loops. This is a consequence of the Mostow rigidity theorem, which states that any two hyperbolic manifolds with the same fundamental group are isometric. Therefore, for hyperbolic manifolds, the universal cover is isometric to the manifold itself, and there is a unique closed geodesic in each free homotopy class of loops.

For other manifolds occurring in Thurston's geometrization theorem, the situation is more complicated. In general, it is not known whether there is a unique closed geodesic in each free homotopy class of loops for these manifolds. However, there are some conditions under which this is true. For example, if the manifold is geometrically finite, then there is a unique closed geodesic in each free homotopy class of loops. This was proven by Marden in 1974.

2. When a homotopy class does not have a unique closed geodesic, can anything be said about the number of closed geodesics in that free homotopy class?

For manifolds that do not have a unique closed geodesic in each free homotopy class, it is possible to have infinitely many closed geodesics in a given homotopy class. This is the case for certain flat manifolds. However, for other manifolds, such as hyperbolic manifolds, it is conjectured that there are only finitely many closed geodesics in each free homotopy class. This is known as the Lehmer conjecture.

3. Is there any easily stated theorem describing the structure of closed geodesics in three manifolds?

There are several theorems that describe the structure of closed geodesics in three manifolds. One such theorem is the Bonnet-Myers theorem, which states that if a three-dimensional Riemannian manifold has positive sectional curvature everywhere, then it is

 

[tacman]

Thank you for your interest in this topic. I will do my best to address your questions and provide some additional information on the structure of closed geodesics in three manifolds.

1. Conditions for unique closed geodesic in each free homotopy class of loops
As you have correctly stated, for hyperbolic manifolds, there is a unique closed geodesic in each free homotopy class of loops. This is a consequence of the fact that hyperbolic manifolds have negative curvature and satisfy the property of having "thin" geodesic flow. This means that the geodesic flow on a hyperbolic manifold is ergodic, which in turn implies that every free homotopy class of loops contains a unique closed geodesic.

For other manifolds occurring in Thurston's geometrization theorem, such as spherical and flat manifolds, there may not be a unique closed geodesic in each free homotopy class of loops. This is because these manifolds have positive or zero curvature, which can lead to multiple closed geodesics in the same homotopy class. However, there are still conditions under which a unique closed geodesic can be guaranteed. For example, if the manifold has a non-trivial isometry group, then there will be a unique closed geodesic in each free homotopy class of loops that is invariant under this group.

2. Number of closed geodesics in a homotopy class
In general, it is difficult to say anything definitive about the number of closed geodesics in a given homotopy class. This is because the number of closed geodesics can vary greatly depending on the specific manifold and the homotopy class in question. For example, some homotopy classes may have infinitely many closed geodesics, while others may only have a finite number. Additionally, the lengths of these geodesics may or may not be related in any meaningful way.

3. Theorems and conjectures on closed geodesics in three manifolds
There are several theorems and conjectures that describe the structure of closed geodesics in three manifolds. One such theorem is the "prime geodesic theorem," which states that every prime closed geodesic in a compact three-manifold is simple, meaning it does not intersect itself. This theorem has been proven for hyperbolic and spherical manifolds