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Isochronous centers and flat Finsler metrics (I)

Part of: Global differential geometry Low-dimensional dynamical systems

Published online by Cambridge University Press:  15 April 2024

Xinhe Mu ,
Hui Miao  and
Libing Huang Open the ORCID record for Libing Huang [Opens in a new window]
Xinhe Mu
Affiliation:
Academy of Mathematics and System Sciences, CAS, Beijing, China e-mail: muxinhe22@mails.ucas.ac.cn
Hui Miao
Affiliation:
School of Mathematical Sciences, Nankai University, Tianjin, China e-mail: 2805582641@qq.com
Libing Huang*
Affiliation:
School of Mathematical Sciences and LPMC, Nankai University, Tianjin, China
*
e-mail: huanglb@nankai.edu.cn
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Abstract

The local structure of rotationally symmetric Finsler surfaces with vanishing flag curvature is completely determined in this paper. A geometric method for constructing such surfaces is introduced. The construction begins with a planar vector field X that depends on two functions of one variable. It is shown that the flow of X could be used to generate a generalized Finsler surface with zero flag curvature. Moreover, this generalized structure reduces to a regular Finsler metric if and only if X has an isochronous center. By relating X to a Liénard system, we obtain the isochronicity condition and discover numerous new examples of complete flat Finsler surfaces, depending on an odd function and an even function.

Keywords

Isochronous centerLiénard systemflag curvatureFinsler metric

MSC classification

Primary: 53C60: Finsler spaces and generalizations (areal metrics)
Secondary: 37E35: Flows on surfaces
Type
Article
Information
Canadian Journal of Mathematics , First View , pp. 1 - 21
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Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of Canadian Mathematical Society

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Footnotes

This work is supported by the National Natural Science Foundation of China (Grant No. 12131012).

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