A systolic inequality for geodesic flows on the two-sphere

作者：Abbondandolo Alberto; Bramham Barney; Hryniewicz Umberto L; Salomao Pedro A S

来源：Mathematische Annalen, 2017, 367(1-2): 701-753.

DOI：10.1007/s00208-016-1385-2

For a Riemannian metric g on the two-sphere, let be the length of the shortest closed geodesic and 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) <= pi 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.

• 出版日期2017-2