Geodesic behavior for Finsler metrics of constant positive flag curvature on S2

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Geodesic behavior for Finsler metrics of constant positive flag curvature on S². / Bryant, R. L.; Foulon, P.; Ivanov, S. V.; Matveev, V. S.; Ziller, W.

In: Journal of Differential Geometry, Vol. 117, No. 1, 01.2021, p. 1-22.

Research output: Contribution to journal › Article › peer-review

Author

Bryant, R. L. ; Foulon, P. ; Ivanov, S. V. ; Matveev, V. S. ; Ziller, W. / Geodesic behavior for Finsler metrics of constant positive flag curvature on S². In: Journal of Differential Geometry. 2021 ; Vol. 117, No. 1. pp. 1-22.

BibTeX

@article{1f98b29c036b42a8b0380c0d6af19f55,

title = "Geodesic behavior for Finsler metrics of constant positive flag curvature on S2",

abstract = "We study non-reversible Finsler metrics with constant flag curvature 1 on S2 and show that the geodesic flow of every such metric is conjugate to that of one of Katok's examples, which form a 1- parameter family. In particular, the length of the shortest closed geodesic is a complete invariant of the geodesic flow. We also show, in any dimension, that the geodesic flow of a Finsler metric with constant positive flag curvature is completely integrable. Finally, we give an example of a Finsler metric on S2with positive flag curvature such that no two closed geodesics intersect and show that this is not possible when the metric is reversible or has constant flag curvature.",

keywords = "NAVIGATION, MANIFOLDS",

author = "Bryant, {R. L.} and P. Foulon and Ivanov, {S. V.} and Matveev, {V. S.} and W. Ziller",

note = "Funding Information: R. Bryant thanks Duke University for a research grant and the U.S. National Science Foundation for the grant DMS-0103884, S. Ivanov was supported by the RFBR grant 17-01-00128, V. Matveev by the University of Jena and the DFG grant MA 2565/4, and W. Ziller by the NSF grant DMS-1506148. Funding Information: Acknowledgments. The authors thank David Bao, Alexey Bolsinov, Sergei Matveev, Ioan Radu Peter, Jean-Philippe Pr{\'e}aux, Hans-Bert Rademacher, Sorin Sabau, Zhongmin Shen, and Oksana Yakimova for useful comments and help in finding appropriate references. Some of our results were obtained during the conference “New Methods in Finsler Geometry”, which took place in July 2016 in Leipzig and was supported by the DFG and the Universities of Jena and Leipzig. Publisher Copyright: {\textcopyright} 2021 International Press of Boston, Inc.. All rights reserved. Copyright: Copyright 2021 Elsevier B.V., All rights reserved.",

year = "2021",

month = jan,

doi = "10.4310/JDG/1609902015",

language = "English",

volume = "117",

pages = "1--22",

journal = "Journal of Differential Geometry",

issn = "0022-040X",

publisher = "International Press of Boston, Inc.",

number = "1",

}

RIS

TY - JOUR

T1 - Geodesic behavior for Finsler metrics of constant positive flag curvature on S2

AU - Bryant, R. L.

AU - Foulon, P.

AU - Ivanov, S. V.

AU - Matveev, V. S.

AU - Ziller, W.

N1 - Funding Information: R. Bryant thanks Duke University for a research grant and the U.S. National Science Foundation for the grant DMS-0103884, S. Ivanov was supported by the RFBR grant 17-01-00128, V. Matveev by the University of Jena and the DFG grant MA 2565/4, and W. Ziller by the NSF grant DMS-1506148. Funding Information: Acknowledgments. The authors thank David Bao, Alexey Bolsinov, Sergei Matveev, Ioan Radu Peter, Jean-Philippe Préaux, Hans-Bert Rademacher, Sorin Sabau, Zhongmin Shen, and Oksana Yakimova for useful comments and help in finding appropriate references. Some of our results were obtained during the conference “New Methods in Finsler Geometry”, which took place in July 2016 in Leipzig and was supported by the DFG and the Universities of Jena and Leipzig. Publisher Copyright: © 2021 International Press of Boston, Inc.. All rights reserved. Copyright: Copyright 2021 Elsevier B.V., All rights reserved.

PY - 2021/1

Y1 - 2021/1

N2 - We study non-reversible Finsler metrics with constant flag curvature 1 on S2 and show that the geodesic flow of every such metric is conjugate to that of one of Katok's examples, which form a 1- parameter family. In particular, the length of the shortest closed geodesic is a complete invariant of the geodesic flow. We also show, in any dimension, that the geodesic flow of a Finsler metric with constant positive flag curvature is completely integrable. Finally, we give an example of a Finsler metric on S2with positive flag curvature such that no two closed geodesics intersect and show that this is not possible when the metric is reversible or has constant flag curvature.

AB - We study non-reversible Finsler metrics with constant flag curvature 1 on S2 and show that the geodesic flow of every such metric is conjugate to that of one of Katok's examples, which form a 1- parameter family. In particular, the length of the shortest closed geodesic is a complete invariant of the geodesic flow. We also show, in any dimension, that the geodesic flow of a Finsler metric with constant positive flag curvature is completely integrable. Finally, we give an example of a Finsler metric on S2with positive flag curvature such that no two closed geodesics intersect and show that this is not possible when the metric is reversible or has constant flag curvature.

KW - NAVIGATION

KW - MANIFOLDS

UR - http://www.scopus.com/inward/record.url?scp=85100204698&partnerID=8YFLogxK

U2 - 10.4310/JDG/1609902015

DO - 10.4310/JDG/1609902015

M3 - Article

AN - SCOPUS:85100204698

VL - 117

SP - 1

EP - 22

JO - Journal of Differential Geometry

JF - Journal of Differential Geometry

SN - 0022-040X

IS - 1

ER -

ID: 75047811