Duality of Singular Paths for (2, 3, 5)-Distributions
- 169 Downloads
We show a duality which arises from distributions of Cartan type, having growth (2, 3, 5), from the viewpoint of geometric control theory. In fact, we consider the space of singular (or abnormal) paths on a given five-dimensional space endowed with a Cartan distribution, which form another five-dimensional space with a cone structure. We regard the cone structure as a control system and show that the space of singular paths of the cone structure is naturally identified with the original space. Moreover, we observe an asymmetry on this duality in terms of singular paths.
KeywordsSingular control Cartan prolongation Cone structure
Mathematics Subject Classifications (2010)58A30 53A55 53C17
This work was supported by KAKENHI no. 22340030 and no. 23654058.
- 1.Agrachev AA. Geometry of optimal control problems and Hamiltonian systems. In: Nonlinear and optimal control theory. Berlin: Springer; 2004. p. 1–59.Google Scholar
- 3.Agrachev A, Zelenko I. Nurowski’s conformal structures for (2, 5)-distributions via dynamics of abnormal extremals. In: Proceedings of RIMS symposium on developments of cartan geometry and related mathematical problems, RIMS Kokyuroku. 2006. vol. 1502, p. 204–218. arXiv:math.DG/0.605059.
- 4.Bonnard B, Chyba M. Singular trajectories and the role in control theory. Berlin: Springer; 2003.Google Scholar
- 5.Bryant RL. Élie Cartan and geometric duality. A lecture given at the Institut d’Élie Cartan on 19 June; 1998.Google Scholar
- 10.Doubrov B, Zelenko I. Prolongation of quasi-principal frame bundles and geometry of flag structures on manifolds. preprint, arXiv:math.DG/1210.7334v2.
- 13.Ishikawa G, Machida Y, Takahashi M. Singularities of tangent surfaces in Cartan’s split G 2-geometry. Hokkaido University Preprint Series in Mathematics #1020; 2012.Google Scholar
- 14.Kitagawa Y. The infinitesimal automorphisms of a homogeneous subriemannian contact manifold. Thesis, Nara Women’s University; 2005.Google Scholar
- 15.Liu W, Sussman HJ. Shortest paths for sub-Riemannian metrics on rank-two distributions. Memoirs of American mathematical society, vol. 118–564. American Mathematical Society; 1995.Google Scholar
- 16.Montgomery R. A tour of Subriemannian geometries, their geodesics and applications. Mathematical surveys and monographs, vol. 91. Am Math Soc. 2002.Google Scholar
- 23.Zhitomirskii M. Exact normal form for (2, 5) distributions. RIMS Kokyuroku. 2006;1502:16–28.Google Scholar