Sharp systolic inequalities for Riemannian and Finsler spheres of revolution

Abbondandolo, Alberto; Bramham, Barney; Hryniewicz, Umberto; Salomão, Pedro

doi:10.1090/tran/8233

Sharp systolic inequalities for Riemannian and Finsler spheres of revolution
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by Alberto Abbondandolo, Barney Bramham, Umberto L. Hryniewicz and Pedro A. S. Salomão PDF

Trans. Amer. Math. Soc. 374 (2021), 1815-1845 Request permission

Abstract:

We prove that the systolic ratio of a sphere of revolution $S$ does not exceed $\pi$ and equals $\pi$ if and only if $S$ is Zoll. More generally, we consider the rotationally symmetric Finsler metrics on a sphere of revolution which are defined by shifting the tangent unit circles by a Killing vector field. We prove that in this class of metrics the systolic ratio does not exceed $\pi$ and equals $\pi$ if and only if the metric is Riemannian and Zoll.

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