Hostname: page-component-78c5997874-8bhkd Total loading time: 0 Render date: 2024-11-12T19:45:10.877Z Has data issue: false hasContentIssue false
  • English
  • Français

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 ([email protected], [email protected])
Thomas Mettler
Affiliation:
Institut für Mathematik, Goethe-Universität Frankfurt, 60325 Frankfurt am Main, Germany ([email protected], [email protected])
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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
Check for updates
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

References

Adem, A., Leida, J. and Ruan, Y., Orbifolds and Stringy Topology Cambridge Tracts in Mathematics 171 (Cambridge University Press, Cambridge, UK, 2007).Google Scholar
Akbar-Zadeh, H., Sur les espaces de Finsler à courbures sectionnelles constantes, Acad. Roy. Belg. Bull. Cl. Sci. (5) 74 (1988), 281–322.Google Scholar
Amann, M., Lange, C. and Radeschi, M., ‘Odd-dimensional orbifolds with all geodesics closed are covered by manifolds’, Preprint, 2018, https://arxiv.org/abs/1811.10320.Google Scholar
Besse, A. L., Manifolds All of Whose Geodesics Are Closed, Ergebnisse der Mathematik und ihrer Grenzgebiete, 93 (Springer-Verlag, Berlin, 1978). With appendices by Epstein, D. B. A., Bourguignon, J.-P., Bérard-Bergery, L., Berger, M. and Kazdan, J. L..CrossRefGoogle Scholar
Bridson, M. R. and Haefliger, A., Metric Spaces of Non-Positive Curvature, Grundlehren der Mathematischen Wissenschaften, 319 (Springer-Verlag, Berlin, 1999).CrossRefGoogle Scholar
Bryant, R. L., Finsler Surfaces with Prescribed Curvature Conditions, Unpublished manuscript, 1995.Google Scholar
Bryant, R. L., Finsler structures on the $2$ -sphere satisfying $K=1$ , in Finsler Geometry (Seattle, WA, 1995), Contemporary Mathematics, 196, pp. 27–41 (American Mathematical Society, Providence, RI, 1996).CrossRefGoogle Scholar
Bryant, R. L., Projectively flat Finsler $2$ -spheres of constant curvature, Selecta Math. (N.S.) 3 (1997), 161–203.CrossRefGoogle Scholar
Bryant, R. L., Some remarks on Finsler manifolds with constant flag curvature, Houston J. Math. 28 (2002), 221–262. Special issue for Chern, S. S..Google Scholar
Bryant, R. L., Geodesically reversible Finsler 2-spheres of constant curvature, in Inspired by Chern, S. S., Nankai Tracts in Mathematics, 11, pp. 95–111 (World Scientific Publishing, Hackensack, NJ, 2006).Google Scholar
Bryant, R. L., Foulon, P., Ivanov, S. V., Matveev, V. S. and Ziller, W., Geodesic behavior for Finsler metrics of constant positive flag curvature on ${S}^2$ , J. Differential Geom. 117 (2021), 1–22.CrossRefGoogle Scholar
Bryant, R. L., Manno, G. and Matveev, V. S., A solution of a problem of Sophus Lie: normal forms of two-dimensional metrics admitting two projective vector fields, Math. Ann. 340 (2008), 437–463.CrossRefGoogle Scholar
Cartan, E., Sur un problème d’équivalence et la théorie des espaces métriques généralisés., Mathematica 4 (1930), 114–136.Google Scholar
Dubois-Violette, M., Structures complexes au-dessus des variétés, applications, in Mathematics and physics (Paris, 1979/1982), Progress in Mathematics, 37, pp. 1–42 (Birkhäuser Boston, Boston, MA, 1983).Google Scholar
Epstein, D. B. A., Periodic flows on three-manifolds, Ann. of Math. (2) 95 (1972), 66–82.CrossRefGoogle Scholar
Foulon, P., Curvature and global rigidity in Finsler manifolds, Houston J. Math. 28 (2002), 263–292. Special issue for Chern, S. S..Google Scholar
Geiges, H. and Gonzalo-Perez, J., Transversely holomorphic flows and contact circles on spherical 3-manifolds, Enseign. Math. 62 (2016), 527–567.CrossRefGoogle Scholar
Geiges, H. and Lange, C., Seifert fibrations of lens spaces, Abh. Math. Semin. Univ. Hambg. 88 (2018), 1–22.CrossRefGoogle Scholar
Hitchin, N. J., Complex manifolds and Einstein’s equations, in Twistor Geometry and Nonlinear Systems (Primorsko, 1980), Lecture Notes in Mathematics, 970, pp. 73–99 (Springer, Berlin, 1982).CrossRefGoogle Scholar
Imayoshi, Y. and Taniguchi, M., An Introduction to Teichmüller Spaces (Springer-Verlag, Tokyo, 1992). Translated and revised from the Japanese by the authors.CrossRefGoogle Scholar
Ivey, T. A. and Landsberg, J. M., Cartan for Beginners, 2nd ed., Graduate Studies in Mathematics, 175 (American Mathematical Society, Providence, RI, 2016).CrossRefGoogle Scholar
Katok, A. B., Ergodic perturbations of degenerate integrable Hamiltonian systems, Izv. Akad. Nauk SSSR Ser. Mat. 37 (1973), 539–576.Google Scholar
Kobayashi, S., Transformation Groups in Differential Geometry, Classics in Mathematics (Springer-Verlag, Berlin, 1995). Reprint of the 1972 edition.Google Scholar
Lange, C., On metrics on 2-orbifolds all of whose geodesics are closed, J. Reine Angew. Math. 758 (2020), 67–94.CrossRefGoogle Scholar
Lange, C., Orbifolds from a metric viewpoint, Geom. Dedicata 209 (2020), 43–57.CrossRefGoogle Scholar
LeBrun, C. R., Spaces of Complex Geodesics and Related Structures, Ph.D. thesis, University of Oxford, 1980, https://ora.ox.ac.uk/objects/uuid:e29dd99c-0437-4956-8280-89dda76fa3f8.Google Scholar
LeBrun, C. R. and Mason, L. J., Zoll manifolds and complex surfaces, J. Differential Geom. 61 (2002), 453–535.CrossRefGoogle Scholar
LeBrun, C. R. and Mason, L. J., Zoll metrics, branched covers, and holomorphic disks, Comm. Anal. Geom. 18 (2010), 475–502.CrossRefGoogle Scholar
Matveev, V. S., Lichnerowicz-Obata conjecture in dimension two, Comment. Math. Helv. 80 (2005), 541–570.CrossRefGoogle Scholar
Matveev, V. S., Proof of the projective Lichnerowicz-Obata conjecture, J. Differential Geom. 75 (2007), 459–502.CrossRefGoogle Scholar
Mettler, T., Weyl metrisability of two-dimensional projective structures, Math. Proc. Cambridge Philos. Soc. 156 (2014), 99–113.CrossRefGoogle Scholar
Mettler, T., Geodesic rigidity of conformal connections on surfaces, Math. Z. 281 (2015), 379–393.CrossRefGoogle Scholar
Mettler, T. and Paternain, G. P., Holomorphic differentials, thermostats and Anosov flows, Math. Ann. 373 (2019), 553–580.CrossRefGoogle Scholar
Mettler, T. and Paternain, G. P., Convex projective surfaces with compatible Weyl connection are hyperbolic, Anal. PDE 13 (2020), 1073–1097.CrossRefGoogle Scholar
O'Brian, N. R. and Rawnsley, J. H., Twistor spaces, Ann. Global Anal. Geom. 3 (1985), 29–58.CrossRefGoogle Scholar
Satake, I., The Gauss-Bonnet theorem for $V$ -manifolds, J. Math. Soc. Japan 9 (1957), 464–492.CrossRefGoogle Scholar
Scott, P., The geometries of $3$ -manifolds, Bull. Lond. Math. Soc. 15 (1983), 401–487.CrossRefGoogle Scholar
Seifert, H., Topologie Dreidimensionaler Gefaserter Räume, Acta Math. 60 (1933), 147–238.CrossRefGoogle Scholar
Thurston, W. P., The geometry and topology of three-manifolds, Lecture notes, 1979, Princeton University.Google Scholar
Zhu, S., Critical metrics on $2$ -dimensional orbifolds, Indiana Univ. Math. J. 46 (1997), 1273–1288.CrossRefGoogle Scholar
Ziller, W., Geometry of the Katok examples, Ergodic Theory Dynam. Systems 3 (1983), 135–157.CrossRefGoogle Scholar
Zoll, O., Ūber Flächen mit Scharen geschlossener geodätischer Linien, Math. Ann. 57 (1903), 108–133.CrossRefGoogle Scholar