141 research outputs found
The Length of a Shortest Geodesic Loop
We give a lower bound for the length of a non-trivial geodesic loop on a
simply-connected and compact manifold of even dimension with a non-reversible
Finsler metric of positive flag curvature. Harris and Paternain use this
estimate in their recent paper [HP] to give a geometric characterization of
dynamically convex Finsler metrics on the 2-sphere.Comment: 4 page
Closed geodesics on connected sums and 3-manifolds
We study the asymptotics of the number N(t) of geometrically distinct closed
geodesics of a Riemannian or Finsler metric on a connected sum of two compact
manifolds of dimension at least three with non-trivial fundamental groups and
apply this result to the prime decomposition of a three-manifold. In particular
we show that the function N(t) grows at least like the prime numbers on a
compact 3-manifold with infinite fundamental group. It follows that a generic
Riemannian metric on a compact 3-manifold has infinitely many geometrically
distinct closed geodesics. We also consider the case of a connected sum of a
compact manifold with positive first Betti number and a simply-connected
manifold which is not homeomorphic to a sphere.Comment: 15 page
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