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

Joseph E. Borzellino·Christopher R. Jordan-Squire·Gregory C. Petrics·D. Mark Sullivan
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Abstract

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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Joseph E. Borzellino, Christopher R. Jordan-Squire, Gregory C. Petrics, D. Mark Sullivan. 2006-02-27. On the existence of infinitely many closed geodesics on orbifolds of revolution. https://arxiv.org/abs/math/0602595

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