Abstract
Weyl (Zur Infinitisimalgeometrie: Einordnung der projektiven und der konformen Auffasung, Nachrichten von der K. Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse, Göttinger Akademie der Wissenschaften, Göttingen, 1921) demonstrated that for a connected manifold of dimension greater than 1, if two Riemannian metrics are conformal and have the same geodesics up to a reparametrization, then one metric is a constant scaling of the other one. In the present paper we investigate the analogous property for sub-Riemannian metrics. In particular, we prove that the analogous statement, called the Weyl rigidity, holds either in real analytic category for all sub-Riemannian metrics on distributions with a specific property of their complex abnormal extremals, called minimal order, or in smooth category for all distributions such that all complex abnormal extremals of their nilpotent approximations are of minimal order. This also shows, in real analytic category, the genericity of distributions for which all sub-Riemannian metrics are Weyl rigid and genericity of Weyl rigid sub-Riemannian metrics on a given bracket generating distributions. Finally, this allows us to get analogous genericity results for projective rigidity of sub-Riemannian metrics, i.e., when the only sub-Riemannian metric having the same sub-Riemannian geodesics, up to a reparametrization, with a given one, is a constant scaling of this given one. This is the improvement of our results on the genericity of weaker rigidity properties proved in recent paper (Jean et al. in Geom Dedic 203(1):279–319, 2019).
Similar content being viewed by others
Notes
In fact in the original formulation in [6] we used the term ample instead of generic, see [Definition 2.9] there, but we do not really need this technicalities here.
From now on to simplify the notation in all relation involving functions on open sets of \(T^*M\) \(\alpha \) actually will mean \(\alpha \circ \pi \).
Here we could impose a weaker condition \(d\delta _1\ne 0\) but we will need the given stronger condition in the next section.
References
Agrachev, A., Barilari, D., Rizzi, L.: Curvature: a variational approach. Mem. AMS 256, 1225 (2018)
Chitour, Y., Jean, F., Trélat, E.: Genericity results for singular curves. J. Differ. Geom. 73(1), 45–73 (2006)
Chitour, Y., Jean, F., Trélat, E.: Singular trajectories of control-affine systems. SIAM J. Control Optim. 47(2), 1078–1095 (2008)
de Jong, T., Pfister, G.: Local Analytic Geometry: Basic Theory and Applications, Advanced Lectures in Mathematics. Friedr. Vieweg & Sohn, Braunschweig (2000)
Doubrov, B., Zelenko, I.: On local geometry of vector distribution with given Jacobi symbols. Preprint, submitted. arXiv:1610.09577 [math.DG]
Jean, F., Maslovskaya, S., Zelenko, I.: On projective and affine equivalence of sub-Riemannian metrics. Geom. Dedic. 203(1), 279–319 (2019)
Levi-Civita, T.: Sulle trasformazioni delle equazioni dinamiche. Ann. Mat. Ser. 2a 24, 255–300 (1896)
Weyl, H.: Zur Infinitisimalgeometrie: Einordnung der projektiven und der konformen Auffasung, Nachrichten von der K. Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse, Göttinger Akademie der Wissenschaften, Göttingen (1921); Selecta Hermann Weyl, Birkäuser Verlag, Basel und Stuttgart (1956)
Zhitomirskii, M.: Typical Singularities of Differential 1-Forms and Pfaffian Equations. Translated from Russian, Translations of Mathematical Monograph, 113, American Mathematical Society, Providence, RI; in cooperation with Mir Publisher, Moscow (1992)
Zelenko, I.: On the geodesic equivalence of Riemannian metrics and sub-Riemannian metrics on distributions of corank 1, Sovrem. Mat. Prilozh. No. 21, Geom. Zadachi Teor. Upr., 79–105, 2004; Engl.transl.: J. Math. Sci. (N. Y.) 135 (4), 3168–3194 (2006)
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
This work was supported by a public grant as part of the Investissement d’avenir project, reference ANR-11-LABX-0056-LMH, LabEx LMH, in a joint call with Programme Gaspard Monge en Optimisation et Recherche Opérationnelle, by the iCODE Institute project funded by the IDEX Paris-Saclay, ANR-11-IDEX-0003-02 and by the Grant ANR-15-CE40-0018 of the ANR. I. Zelenko was partly supported by NSF Grant DMS-1406193 and Simons Foundation Collaboration Grant for Mathematicians 524213.
Rights and permissions
About this article
Cite this article
Jean, F., Maslovskaya, S. & Zelenko, I. On Weyl’s type theorems and genericity of projective rigidity in sub-Riemannian geometry. Geom Dedicata 213, 295–314 (2021). https://doi.org/10.1007/s10711-020-00581-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10711-020-00581-z
Keywords
- Sub-Riemannian geometry
- Riemannian geometry
- Conformal geometry
- Projective geometry
- Normal geodesics
- Abnormal geodesics
- Nilpotent approximation