Skip to main content Accessibility help


  • Sylvain Arguillère (a1) and Emmanuel Trélat (a2)


In this paper, we define and study strong right-invariant sub-Riemannian structures on the group of diffeomorphisms of a manifold with bounded geometry. We derive the Hamiltonian geodesic equations for such structures, and we provide examples of normal and of abnormal geodesics in that infinite-dimensional context. The momentum formulation gives a sub-Riemannian version of the Euler–Arnol’d equation. Finally, we establish some approximate and exact reachability properties for diffeomorphisms, and we give some consequences for Moser theorems.



Hide All
1. Agrachev, A. A., Any sub-Riemannian metric has points of smoothness, Russian Math. Dokl. 79 (2009), 13.
2. Agrachev, A. A., Boscain, U., Charlot, G., Ghezzi, R. and Sigalotti, M., Two-dimensional almost-Riemannian structures with tangency points, Ann. Inst. H. Poincaré Anal. Non Linéaire 27(3) (2010), 793807.
3. Agrachev, A. A. and Caponigro, M., Controllability on the group of diffeomorphisms, Ann. Inst. H. Poincaré Anal. Non Linéaire 26(6) (2009), 25032509.
4. Ambrosio, L., Transport equation and Cauchy problem for BV vector fields, Invent. Math. 158(2) (2004), 227260.
5. Arguillère, S., Trélat, E., Trouvé, A. and Younes, and L., Shape deformation analysis from the optimal control viewpoint, J. Math. Pures Appl. 104 (2015), 139178.
6. Arnol’d, V., Sur la géométrie différentielle des groupes de Lie de dimension infinie et ses applications à l’hydrodynamique des fluides parfaits, Ann. Inst. Fourier (Grenoble) 16(1) (1966), 319361.
7. Bauer, M., Bruveris, M., Harms, P. and Michor, P. W., Geodesic distance for right invariant sobolev metrics of fractional order on the diffeomorphism group, Ann. Global Anal. Geom. 44(1) (2013), 521.
8. Bauer, M., Bruveris, M. and Michor, P. W., Overview of the geometries of shape spaces and diffeomorphism groups, J. Math. Imaging Vision 50 (2014), 6097.
9. Bauer, M., Harms, P. and Michor, P. W., Sobolev metrics on the manifold of all Riemannian metrics, J. Differential Geom. 94(2) (2013), 187208.
10. Bauer, M., Harms, P. and Michor, P. W., Sobolev metrics on shape space of surfaces, J. Geom. Mech. 3(4) (2011), 389438.
11. Bellaïche, A., The tangent space in sub-Riemannian geometry, in Sub-Riemannian Geometry, Progress in Mathematics, Volume 144, pp. 178 (Birkhäuser, Basel, 1996).
12. Boscain, U., Duplaix, J., Gauthier, J. P. and Rossi, F., Anthropomorphic image reconstruction via hypoelliptic diffusion, SIAM J. Control Optim. 50(3) (2012), 13091336.
13. Boscain, U., Gauthier, J. P. and Rossi, F., Hypoelliptic heat kernel over 3-step nilpotent Lie groups, J. Math. Sci. 199(6) (2014), 614628.
14. Bruveris, M. and Vialard, F.-X., On completeness of groups of diffeomorphisms, preprint, 2014.
15. Chitour, Y., Jean, F. and Trélat, E., Genericity results for singular curves, J. Differential Geom. 73(1) (2006), 4573.
16. Chitour, Y., Jean, F. and Trélat, E., Singular trajectories of control-affine systems, SIAM J. Control Optim. 47(2) (2008), 10781095.
17. DiPerna, R. J. and Lions, P.-L., Ordinary differential equations, transport theory and Sobolev spaces, Invent. Math. 98(3) (1989), 511547.
18. Dudnikov, P. I. and Samborski, S. N., Controllability criterion for systems in a Banach space (generalization of Chow’s theorem), Ukrain. Mat. Zh. 32(5) (1980), 649653.
19. Dupuis, P., Grenander, U. and Miller, M. I., Variational problems on flows of diffeomorphisms for image matching, Quart. Appl. Math. 56(3) (1998), 587600.
20. Ebin, D. G. and Marsden, J., Groups of diffeomorphisms and the motion of an incompressible fluid, Ann. of Math. (2) 92 (1970), 102163.
21. Eichhorn, J., Global Analysis on Open Manifolds (Nova Science Publishers Inc., 2007).
22. Eichhorn, J. and Schmid, R., Form preserving diffeomorphisms on open manifolds, Ann. Global Anal. Geom. 14(2) (1996), 147176.
23. Figalli, A. and Rifford, L., Mass transportation on sub-Riemannian manifolds, Geom. Funct. Anal. 20(1) (2010), 124159.
24. Grenander, U. and Miller, M. I., Computational anatomy: an emerging discipline, Quart. Appl. Math. 56(4) (1998), 617694; Current and future challenges in the applications of mathematics (Providence, RI, 1997).
25. Grong, E., Markina, I. and Vasil’ev, A., Sub-riemannian geometry on infinite-dimensional manifolds, J. Geom. Anal. (2012) (to appear), arXiv:1201.2251.
26. Heintze, E. and Liu, X., Homogeneity of infinite dimensional isoparametric submanifolds, Ann. of Math. (2) 149(1) (1999), 149181.
27. Holm, D., Marsden, J. E. and Ratiu., T. S., The Euler-Poincaré equations and semidirect products with applications to continuum theories, Adv. Math. 137 (1998), 181.
28. Khesin, B. and Lee, P., A nonholonomic Moser theorem and optimal transport, J. Symplectic Geom. 7(4) (2009), 381414.
29. Kriegl, A. and Michor, P. W., The Convenient Setting of Global Analysis, Mathematical Surveys and Monographs, Volume 53 (American Mathematical Society, Providence, RI, 1997).
30. Kurcyusz., S., On the existence and nonexistence of Lagrange multipliers in Banach spaces, J. Optim. Theory Appl. 20(1) (1976), 81110.
31. Li, X. J. and Yong, J. M., Optimal control theory for infinite-dimensional systems, in Systems & Control: Foundations & Applications (Birkhäuser Boston Inc., Boston, MA, 1995).
32. Marsden, J. E. and Ratiu, T. S., Introduction to mechanics and symmetry, in A Basic Exposition of Classical Mechanical Systems, 2nd ed., Texts in Applied Mathematics, Volume 17 (Springer-Verlag, New York, 1999). xviii+582 pp.
33. Michor, P. W. and Mumford, D., An overview of the Riemannian metrics on spaces of curves using the Hamiltonian approach, Appl. Comput. Harmon. Anal. 23(1) (2007), 74113.
34. Montgomery, R., A Tour of Subriemannian Geometries, Their Geodesics and Applications, Mathematical Surveys and Monographs, Volume 91 (American Mathematical Society, Providence, RI, 2002).
35. Moser, J., On the volume elements on a manifold, Trans. Amer. Math. Soc. 120 (1965), 286294.
36. Omori, H., Infinite Dimensional Lie Transformation Groups, Lecture Notes in Mathematics, Volume 427 (Springer-Verlag, Berlin, 1974).
37. Pontryagin, L. S., Boltyanskii, V. G., Gamkrelidze, R. V. and Mishchenko, E. F., The mathematical theory of optimal processes, in A Pergamon Press Book (The Macmillan Co., New York, 1964).
38. Rifford, L. and Trélat, E., Morse-Sard type results in sub-Riemannian geometry, Math. Ann. 332(1) (2005), 145159.
39. Salehani, M. K. and Markina, I., Controllability on infinite-dimensional manifolds: a Chow–Rashevsky theorem, Acta Appl. Math. 134(1) (2014), 229246.
40. Schmid, R., Infinite dimensional Lie groups with applications to mathematical physics, J. Geom. Symmetry Phys. 1 (2004), 54120.
41. Trélat, E., Contrôle optimal, théorie & applications (French) [Optimal control, theory and applications]. Vuibert, Paris, 2005.
42. Trouvé, A., Action de groupe de dimension infinie et reconnaissance de formes, C. R. Acad. Sci. Paris I 321(8) (1995), 10311034.
43. Trouvé, A., Diffeomorphism groups and pattern matching in image analysis, Int. J. Comput. Vis. 37(1) (2005), 17.
44. Trouvé, A. and Younes, L., Local geometry of deformable templates, SIAM J. Math. Anal. 37(1) (2005), 1759 (electronic).
45. Trouvé, A. and Younes., L., Shape spaces, in Handbook of Mathematical Methods in Imaging (ed. Scherzer, O.), pp. 13091362 (Springer, New York, 2011).
46. Younes, L., Shapes and Diffeomorphisms, Applied Mathematical Sciences, Volume 171 (Springer-Verlag, Berlin, 2010).
MathJax is a JavaScript display engine for mathematics. For more information see


MSC classification

Related content

Powered by UNSILO


  • Sylvain Arguillère (a1) and Emmanuel Trélat (a2)


Altmetric attention score

Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed.