Geodesic
On Jacobi fields and a canonical connection in sub-Riemannian geometry
Archivum mathematicum, Tome 53 (2017) no. 2, pp. 77-92
Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

Voir la notice de l'article

In sub-Riemannian geometry the coefficients of the Jacobi equation define curvature-like invariants. We show that these coefficients can be interpreted as the curvature of a canonical Ehresmann connection associated to the metric, first introduced in [15]. We show why this connection is naturally nonlinear, and we discuss some of its properties.
In sub-Riemannian geometry the coefficients of the Jacobi equation define curvature-like invariants. We show that these coefficients can be interpreted as the curvature of a canonical Ehresmann connection associated to the metric, first introduced in [15]. We show why this connection is naturally nonlinear, and we discuss some of its properties.
DOI : 10.5817/AM2017-2-77
Classification : 53B15, 53B21, 53C17
Keywords: sub-Riemannian geometry; curvature; connection; Jacobi fields 
@article{10_5817_AM2017_2_77,
     author = {Barilari, Davide and Rizzi, Luca},
     title = {On {Jacobi} fields and a canonical connection in {sub-Riemannian} geometry},
     journal = {Archivum mathematicum},
     pages = {77--92},
     year = {2017},
     volume = {53},
     number = {2},
     doi = {10.5817/AM2017-2-77},
     mrnumber = {3672782},
     zbl = {06770053},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.5817/AM2017-2-77/}
}
TY  - JOUR
AU  - Barilari, Davide
AU  - Rizzi, Luca
TI  - On Jacobi fields and a canonical connection in sub-Riemannian geometry
JO  - Archivum mathematicum
PY  - 2017
SP  - 77
EP  - 92
VL  - 53
IS  - 2
UR  - http://geodesic.mathdoc.fr/articles/10.5817/AM2017-2-77/
DO  - 10.5817/AM2017-2-77
LA  - en
ID  - 10_5817_AM2017_2_77
ER  - 
%0 Journal Article
%A Barilari, Davide
%A Rizzi, Luca
%T On Jacobi fields and a canonical connection in sub-Riemannian geometry
%J Archivum mathematicum
%D 2017
%P 77-92
%V 53
%N 2
%U http://geodesic.mathdoc.fr/articles/10.5817/AM2017-2-77/
%R 10.5817/AM2017-2-77
%G en
%F 10_5817_AM2017_2_77
Barilari, Davide; Rizzi, Luca. On Jacobi fields and a canonical connection in sub-Riemannian geometry. Archivum mathematicum, Tome 53 (2017) no. 2, pp. 77-92. doi: 10.5817/AM2017-2-77

[1] Agrachev, A.A.: Some open problems. Geometric control theory and sub-Riemannian geometry, Springer INdAM Ser., 5, Springer, Cham, 2014, pp. 1–13. | MR | Zbl

[2] Agrachev, A.A., Barilari, D., Boscain, U.: Introduction to Riemannian and sub-Riemannian geometry (Lecture Notes). 2015, http://webusers.imj-prg.fr/~davide.barilari/notes.php

[3] Agrachev, A.A., Barilari, D., Rizzi, L.: Curvature: a variational approach. Memoirs of the AMS (in press).

[4] Agrachev, A.A., Barilari, D., Rizzi, L.: Sub-riemannian curvature in contact geometry. J. Geom. Anal. (2016), 1–43, DOI10.1007/s12220-016-9684-0. | DOI | MR

[5] Agrachev, A.A., Zelenko, I.: Geometry of Jacobi curves. I. J. Dynam. Control Systems 8 (1) (2002), 93–140. DOI:  | DOI | MR | Zbl

[6] Agrachev, A.A., Zelenko, I.: Geometry of Jacobi curves. II. J. Dynam. Control Systems 8 (2) (2002), 167–215, DOI10.1023/A:1015317426164. | DOI | MR | Zbl

[7] Barilari, D., Rizzi, L.: Comparison theorems for conjugate points in sub-Riemannian geometry. ESAIM Control Optim. Calc. Var. 22 (2) (2016), 439–472. | DOI | MR | Zbl

[8] Jean, F.: Control of nonholonomic systems: from sub-Riemannian geometry to motion planning. SpringerBriefs in Mathematics, Springer, Cham, 2014. | MR | Zbl

[9] Kobayashi, S., Nomizu, K.: Foundations of differential geometry. Vol. I. Wiley Classics Library. John Wiley & Sons, Inc., New York, 1996, Reprint of the 1963 original, A Wiley-Interscience Publication. | MR

[10] Li, C., Zelenko, I.: Jacobi equations and comparison theorems for corank 1 sub-Riemannian structures with symmetries. J. Geom. Phys. 61 (4) (2011), 781–807. | DOI | MR | Zbl

[11] Montgomery, R.: Abnormal minimizers. SIAM J. Control Optim. 32 (6) (1994), 1605–1620. | DOI | MR | Zbl

[12] Montgomery, R.: A tour of subriemannian geometries, their geodesics and applications. Mathematical Surveys and Monographs ed., vol. 91, AMS, Providence, RI, 2002. | MR | Zbl

[13] Rifford, L.: Sub-Riemannian geometry and optimal transport. SpringerBriefs in Mathematics, Springer, Cham, 2014. | DOI | MR

[14] Rizzi, L., Silveira, P.: Sub-Riemannian Ricci curvatures and universal diameter bounds for 3-Sasakian manifolds. ArXiv e-prints, Sept. 2015, J. Inst. Math. Jussieu (in press).

[15] Zelenko, I., Li, C.: Differential geometry of curves in Lagrange Grassmannians with given Young diagram. Differential Geom. Appl. 27 (6) (2009), 723–742. | DOI | MR | Zbl

Cité par Sources :