Given a surface S in a 3D contact sub-Riemannian manifold M, we investigate the metric structure induced on S by M, in the sense of length spaces. First, we define a coefficient $$ at characteristic points that determines locally the characteristic foliation of S. Next, we identify some global conditions for the induced distance to be finite. In particular, we prove that the induced distance is finite for surfaces with the topology of a sphere embedded in a tight coorientable distribution, with isolated characteristic points.
Keywords: Contact geometry, sub-Riemannian geometry, length space, Riemannian approximation, Gaussian curvature, Heisenberg group
@article{COCV_2022__28_1_A9_0,
author = {Barilari, Davide and Boscain, Ugo and Cannarsa, Daniele},
title = {On the induced geometry on surfaces in {3D} contact {sub-Riemannian} manifolds},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
year = {2022},
publisher = {EDP-Sciences},
volume = {28},
doi = {10.1051/cocv/2021104},
mrnumber = {4371078},
zbl = {1496.53041},
language = {en},
url = {https://www.numdam.org/articles/10.1051/cocv/2021104/}
}
TY - JOUR AU - Barilari, Davide AU - Boscain, Ugo AU - Cannarsa, Daniele TI - On the induced geometry on surfaces in 3D contact sub-Riemannian manifolds JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2022 VL - 28 PB - EDP-Sciences UR - https://www.numdam.org/articles/10.1051/cocv/2021104/ DO - 10.1051/cocv/2021104 LA - en ID - COCV_2022__28_1_A9_0 ER -
%0 Journal Article %A Barilari, Davide %A Boscain, Ugo %A Cannarsa, Daniele %T On the induced geometry on surfaces in 3D contact sub-Riemannian manifolds %J ESAIM: Control, Optimisation and Calculus of Variations %D 2022 %V 28 %I EDP-Sciences %U https://www.numdam.org/articles/10.1051/cocv/2021104/ %R 10.1051/cocv/2021104 %G en %F COCV_2022__28_1_A9_0
Barilari, Davide; Boscain, Ugo; Cannarsa, Daniele. On the induced geometry on surfaces in 3D contact sub-Riemannian manifolds. ESAIM: Control, Optimisation and Calculus of Variations, Tome 28 (2022), article no. 9. doi: 10.1051/cocv/2021104
[1] , and , A Comprehensive Introduction to Sub-Riemannian Geometry. Cambridge Studies in Advanced Mathematics. Cambridge University Press (2019). | MR | Zbl
[2] , and , A Gauss-Bonnet-like formula on two-dimensional almost-Riemannian manifolds. Discr. Continu. Dyn. Syst. A 20 (2008) 801–822. | MR | Zbl | DOI
[3] , , and , Qualitative Theory of Second Order Dynamic Systems. Israel Program for Scientific Trasnlations Ltd. (1973). | MR | Zbl
[4] , and , Introduction to the Qualitative Theory of Dynamical Systems on Surfaces. Vol. 153 of Translations of Mathematical Monographs. American Mathematical Society (1996). | MR | Zbl | DOI
[5] and , Metric normal and distance function in the Heisenberg group. Math. Zeitsch. 256 (2007) 661–684. | MR | Zbl | DOI
[6] and , The Hessian of the distance from a surface in the Heisenberg group. Ann. Acad. Sci. Fennicae. Math. 33 (2008) 35–63. | MR | Zbl
[7] , Size of characteristic sets and functions with prescribed gradient. J. für die Reine Angew. Math. [Crelle’s Journal] 564 (2003) 63–83. | MR | Zbl
[8] , and , Intrinsic curvature of curves and surfaces and a Gauss–Bonnet theorem in the Heisenberg group. Math. Zeitsch. 287 (2017) 1–38. | MR | Zbl | DOI
[9] , and , Volume of small balls and sub-Riemannian curvature in 3D contact manifolds. J. Symplectic Geom. 18 (2020). | MR | Zbl | DOI
[10] , , and , Stochastic processes on surfaces in three-dimensional contact sub-Riemannian manifolds. Preprint arXiv e-print (2020). | MR | Zbl
[11] , Entrelacements et équations de Pfaff., in IIIe rencontre de géométrie du Schnepfenried, 10-15 mai 1982. Vol. 1: Feuilletages. Géométrie symplectique et de contact. Analyse non standard et applications. Astérisqué 107/108 (1983) 87–161. | MR | Numdam
[12] , Tutorial on the center manifold theorem. Lecture Notes in Mathematics. Springer Verlag, Germany (2007). | MR | Zbl
[13] , and , A course in metric geometry. Vol. 33 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI (2001). | MR | Zbl | DOI
[14] , , and , An introduction to the Heisenberg group and the sub-Riemannian isoperimetric problem. Vol. 259 of Progress in Mathematics. Birkhäuser Verlag, Basel (2007). | MR | Zbl
[15] , and , Non-doubling Ahlfors measures, perimeter measures, and the characterization of the trace spaces of Sobolev functions in Carnot-Carathéodory spaces. Mem. Am. Math. Soc. 182 (2006). | MR | Zbl
[16] , and , Sub-Riemannian calculus on hypersurfaces in Carnot groups. Adv. Math. 215 (2007) 292–378. | MR | Zbl | DOI
[17] , and , Integrability of the sub-Riemannian mean curvature of surfaces in the Heisenberg group. Proc. Am. Math. Soc. 140 (2012). | MR | Zbl | DOI
[18] and , Gauss-Bonnet theorem in sub-Riemannian Heisenberg space . J. Dyn. Control Syst. 22 (2016) 807–820. | MR | Zbl | DOI
[19] , Introductory lectures on contact geometry. Topol. Geometry Manifolds (2003) 81–107. | MR | Zbl | DOI
[20] , An Introduction to Contact Topology, Cambridge Studies in Advanced Mathematics. Cambridge University Press (2008). | MR | Zbl
[21] , Convexité en topologie de contact. Commentarii mathematici Helvetici 66 (1991) 637–677. | MR | Zbl | DOI
[22] , Structures de contact en dimension trois et bifurcations des feuilletages de surfaces. Invent. Math. 141 (2000) 615–689. | MR | Zbl | DOI
[23] , Carnot-Carathéodory spaces seen from within. Edited by and . Progress in Mathematics, vol. 144, Birkhäuser Basel, Basel (1996). | MR | Zbl
[24] , and , Differentiability of the Hartman–Grobman linearization. Discr. Continu. Dyn. Syst. 9 (2003) 979–984. | MR | Zbl | DOI
[25] , On local homeomorphisms of Euclidean space. Boletin de la Sociedad Matematica Mexicana 5 (1960) 220–241. | MR | Zbl
[26] , Control of Nonholonomic Systems: from Sub-Riemannian Geometry to Motion Planning. SpringerBriefs in Mathematics Springer International Publishing (2014). | MR | Zbl
[27] , On surfaces in three dimensional contact manifolds. Can. J. Math. 65 (2013) 621–633. | MR | Zbl | DOI
[28] , A Tour of sub-Riemannian Geometries, Their Geodesics and Applications. Mathematical surveys and monographs. American Mathematical Society (2002). | MR | Zbl
[29] , Minimal surfaces in the Heisenberg group. Geometriae Dedicata 104 (2004) 201–231. | MR | Zbl | DOI
[30] , Differential Equations and Dynamical Systems. Texts in Applied Mathematics. Springer, New York (2012). | Zbl | MR
[31] , Sub-Riemannian Geometry and Optimal Transport. SpringerBriefs in Mathematics. Springer International Publishing (2014). | MR | Zbl | DOI
[32] and , Heat content asymptotics for sub-Riemannian manifolds. J. Math. Pures Appl. 148 (2021) 267–307. | MR | Zbl | DOI
[33] and , On some sub-riemannian objects in hypersurfaces of sub-riemannian manifolds. Bull. Aust. Math. Soc. 70 (2004) 177–198. | MR | Zbl | DOI
[34] , Limit of Gaussian and normal curvatures of surfaces in Riemannian approximation scheme for sub-Riemannian three dimensional manifolds and Gauss-Bonnet theorem. Preprint arXiv e-print (2020).
[35] and , Gauss-Bonnet theorems in the affine group and the group of rigid motions of the Minkowski plane. Sci. China Math. (2020). | MR | Zbl
[36] and , Rigid paths of generic 2-distributions on 3-manifolds. Duke Math. J. 79 (1995). | MR | Zbl | DOI
Cité par Sources :
