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Space of geodesics on Zoll three-spheres. (English) Zbl 1039.53088
Fukaya, Kenji (ed.) et al., Minimal surfaces, geometric analysis and symplectic geometry. Based on the lectures of the workshop and conference, Johns Hopkins University, Baltimore, MD, USA, March 16–21, 1999. Tokyo: Mathematical Society of Japan (ISBN 4-931469-18-3/hbk). Adv. Stud. Pure Math. 34, 237-243 (2002).
A Riemannian manifold \((M,g)\) all of whose geodesics are closed with the same minimal period is a Zoll manifold. In this case, the space of geodesics on \(M\) has a natural manifold structure. More precisely, if we denote by \(U^*M\) the unit cotangent bundle of \(M\), then the space of geodesics is the quotient of \(U^*M\) by the characteristic flow, and carries a natural symplectic structure. In this paper, the author shows that the space of geodesics on a Zoll three-sphere is symplectomorphic to the product of two copies of symplectic two-spheres with the same area.
For the entire collection see [Zbl 0994.00028].
53D05 Symplectic manifolds (general theory)
53C22 Geodesics in global differential geometry