On the existence of infinitely many closed geodesics on orbifolds of revolution

Details On the existence of infinitely many closed geodesics on orbifolds of revolution On the existence of infinitely many closed geodesics on orbifolds of revolution

2006-02-27

arxiv.org/abs/math/0602595v1

Mathematics

Differential Geometry

21 pages, 4 figures; for a PDF version see http://www.calpoly.edu/~jborzell/Publications/publications.html

Scientific paper

Using the theory of geodesics on surfaces of revolution, we introduce the period function. We use this as our main tool in showing that any two-dimensional orbifold of revolution homeomorphic to S^2 must contain an infinite number of geometrically distinct closed geodesics. Since any such orbifold of revolution can be regarded as a topological two-sphere with metric singularities, we will have extended Bangert's theorem on the existence of infinitely many closed geodesics on any smooth Riemannian two-sphere. In addition, we give an example of a two-sphere cone-manifold of revolution which possesses a single closed geodesic, thus showing that Bangert's result does not hold in the wider class of closed surfaces with cone manifold structures.

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