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

  1. Oct 21, 2007 #1
    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
     
    Last edited: Oct 21, 2007
  2. jcsd
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