A systolic inequality for geodesic flows on the two-sphere
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- 作者：Alberto Abbondandolo ; Barney Bramham ; Umberto L. Hryniewicz…
- 刊名：Mathematische Annalen
- 出版年：2017
- 出版时间：February 2017
- 年：2017
- 卷：367
- 期：1-2
- 页码：701-753
- 全文大小：
- 刊物类别：Mathematics and Statistics
- 刊物主题：Mathematics, general;
- 出版者：Springer Berlin Heidelberg
- ISSN：1432-1807
- 卷排序：367
文摘
For a Riemannian metric g on the two-sphere, let \(\ell _{\min }(g)\) be the length of the shortest closed geodesic and \(\ell _{\max }(g)\) 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 $$\begin{aligned} \ell _\mathrm{min}(g)^2 \le \pi \ \mathrm{Area}(S^2,g) \le \ell _{\max }(g)^2, \end{aligned}$$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.References1.Abbondandolo, A., Bramham, B., Hryniewicz, U.L., Salomão, P.A.S.: Sharp systolic inequalities for Reeb flows on the three-sphere (2015). arXiv:1504.05258 [math.SG]2.Aleksandrov, A.D.: Vnutrennyaya Geometriya Vypuklyh Poverhnosteĭ [translated as: Intrinsic geometry of convex surfaces. In: Aleksandrov, A.D. (ed.) 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Salomão41.Fakultät für MathematikRuhr Universität BochumBochumGermany2.Fakultät für MathematikRuhr Universität BochumBochumGermany3.Departamento de Matemática AplicadaUniversidade Federal do Rio de JaneiroRio de JaneiroBrazil4.Departamento de MatemáticaUniversidade de São Paulo, Instituto de Matemática e EstatísticaSão PauloBrazil About this article CrossMark Publisher Name Springer Berlin Heidelberg Print ISSN 0025-5831 Online ISSN 1432-1807 About this journal Reprints and Permissions Article actions Export citation .RIS Papers Reference Manager RefWorks Zotero .ENW EndNote .BIB BibTeX JabRef Mendeley Share article Email Facebook Twitter LinkedIn Cookies We use cookies to improve your experience with our site. More information Accept Over 10 million scientific documents at your fingertips