Abstract
In this note, we show that sub-Riemannian manifolds can contain branching normal minimizing geodesics. This phenomenon occurs if and only if a normal geodesic has a discontinuity in its rank at a non-zero time, which in particular for a strictly normal geodesic means that it contains a non-trivial abnormal subsegment. The simplest example is obtained by gluing the three-dimensional Martinet flat structure with the Heisenberg group in a suitable way. We then use this example to construct more general types of branching.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
A. A. Agrachev, D. Barilari, and U. Boscain, A comprehensive introduction to sub-Riemannian geometry, Cambridge Studies in Advanced Mathematics, Cambridge University Press, (2019).
A. A. Agrachev and Y. L. Sachkov, Control theory from the geometric viewpoint, Encyclopaedia of Mathematical Sciences, vol. 87, Springer-Verlag, Berlin, Control Theory and Optimization, II. (2004)
F. Baudoin and N. Garofalo, Curvature-dimension inequalities and Ricci lower bounds for sub-Riemannian manifolds with transverse symmetries, J. Eur. Math. Soc. (JEMS) 19 (2017), no. 1, 151-219.
Z. M. Balogh, A. Kristály, and K. Sipos, Geometric inequalities on Heisenberg groups, Calc. Var. Partial Differential Equations 57 (2018), no. 2, Art. 61, 41.
Z. M. Balogh, A. Kristály, and K. Sipos, Jacobian determinant inequality on corank 1 Carnot groups with applications, J. Funct. Anal. 277 (2019), no. 12, 108293, 36.
D. Barilari and L. Rizzi, Sub-Riemannian interpolation inequalities, Invent. Math. 215 (2019), no. 3, 977-1038.
E. Hakavuori and E. Le Donne, Non-minimality of corners in subriemannian geometry, Invent. Math. 206 (2016), no. 3, 693-704.
W. Liu and H. J. Sussman, Shortest paths for sub-Riemannian metrics on rank-two distributions, Mem. Amer. Math. Soc. 118 (1995), no. 564, x+104.
E. Milman, The quasi curvature-dimension condition with applications to sub-Riemannian manifolds, 2019.
R. Montgomery, A tour of subriemannian geometries, their geodesics and applications, Mathematical Surveys and Monographs, vol. 91, American Mathematical Society, Providence, RI, 2002.
R. Montgomery, Isoholonomic problems and some applications, Comm. Math. Phys. 128 (1990), no. 3, 565-592.
R. Montgomery, Abnormal minimizers, SIAM J. Control Optim. 32 (1994), no. 6, 1605- 1620.
L. Rifford, Sub-Riemannian geometry and optimal transport, Springer Briefs in Mathematics, Springer, Cham, (2014).
L. Rifford, Singulières minimisantes en géométrie sous-Riemannienne, (2017), pp. Exp. No. 1113, 277-301. Sminaire Bourbaki. Vol. 2015/2016. Exposés 1104-1119.
C. Villani, Inégalités isopérimétriques dans les espaces métriques mesurés [d’après F. Cavalletti & A. Mondino], (2019), pp. Exp. No. 1127, 213-265. Séminaire Bourbaki. Vol. 2016/2017. Exposés 1120-1135.
Acknowledgements
This work was supported by the Grants ANR-15-CE40-0018 and ANR-15-IDEX-02. While this note was in preparation, N. Juillet shared with us an example similar to the one of Sect. 4, in a rank-varying structure on \({\mathbb {R}}^4\). We thank the anonymous referee for carefully reading the paper, and suggesting the simple description of branching in terms of magnetic field presented in Sect. 1.1.
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.
Rights and permissions
About this article
Cite this article
Mietton, T., Rizzi, L. Branching Geodesics in Sub-Riemannian Geometry. Geom. Funct. Anal. 30, 1139–1151 (2020). https://doi.org/10.1007/s00039-020-00539-z
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00039-020-00539-z