A systolic inequality for geodesic flows on the two-sphere

For a Riemannian metric g on the two-sphere, let ℓ min ( g ) be the length of the shortest closed geodesic and ℓ max ( g ) be the length of the longest simple closed geodesic. We prove that if the curvature of g is positive and sufficiently pinched, then the sharp systolic inequalities ℓ min ( g ) 2...

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Bibliographic Details
Published in:Mathematische annalen 2017-02, Vol.367 (1-2), p.701-753
Main Authors: Abbondandolo, Alberto, Bramham, Barney, Hryniewicz, Umberto L., Salomão, Pedro A. S.
Format: Article
Language:English
Subjects:
Curvature
Inequalities
Mathematics
Mathematics and Statistics
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Summary:For a Riemannian metric g on the two-sphere, let ℓ min ( g ) be the length of the shortest closed geodesic and ℓ max ( g ) be the length of the longest simple closed geodesic. We prove that if the curvature of g is positive and sufficiently pinched, then the sharp systolic inequalities ℓ min ( g ) 2 ≤ π Area ( S 2 , g ) ≤ ℓ max ( g ) 2 , hold, and each of these two inequalities is an equality if and only if the metric g is Zoll. The first inequality answers positively a conjecture of Babenko and Balacheff. The proof combines arguments from Riemannian and symplectic geometry.
ISSN:0025-5831
1432-1807
DOI:10.1007/s00208-016-1385-2