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DEFORMATIONS OF THE VERONESE EMBEDDING AND FINSLER $2$-SPHERES OF CONSTANT CURVATURE

Part of: Local differential geometry Calculus on manifolds; nonlinear operators Classical differential geometry

Published online by Cambridge University Press:  20 April 2021

Christian Lange Open the ORCID record for Christian Lange [Opens in a new window]  and
Thomas Mettler Open the ORCID record for Thomas Mettler [Opens in a new window]
Christian Lange
Affiliation:
Mathematisches Institut der Universität München, Theresienstr. 39, D-80333 Munich, Germany (lange@math.lmu.de, clange.math@gmail.com)
Thomas Mettler
Affiliation:
Institut für Mathematik, Goethe-Universität Frankfurt, 60325 Frankfurt am Main, Germany (mettler@math.uni-frankfurt.de, mettler@math.ch)
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Abstract

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We establish a one-to-one correspondence between, on the one hand, Finsler structures on the $2$ -sphere with constant curvature $1$ and all geodesics closed, and on the other hand, Weyl connections on certain spindle orbifolds whose symmetric Ricci curvature is positive definite and whose geodesics are all closed. As an application of our duality result, we show that suitable holomorphic deformations of the Veronese embedding $\mathbb {CP}(a_1,a_2)\rightarrow \mathbb {CP}(a_1,(a_1+a_2)/2,a_2)$ of weighted projective spaces provide examples of Finsler $2$ -spheres of constant curvature whose geodesics are all closed.

Keywords

Finsler structuresconstant curvatureZoll metricsspindle orbifoldsholomorphic curves

MSC classification

Primary: 53B40: Finsler spaces and generalizations (areal metrics)
Secondary: 53A20: Projective differential geometry 58C10: Holomorphic maps
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 21 , Issue 6 , November 2022 , pp. 2103 - 2134
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Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s), 2021. Published by Cambridge University Press

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