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Duality on geodesics of Cartan distributions and sub-Riemannian pseudo-product structures

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Given a five dimensional space endowed with a Cartan distribution, the abnormal geodesics form another five dimensional space with a cone structure. Then it is shown in [15], that, if the cone structure is regarded as a control system, then the space of abnormal geodesics of the cone structure is naturally identified with the original space. In this paper, we provide an exposition on the duality by abnormal geodesics in a wider framework, namely, in terms of quotients of control systems and sub-Riemannian pseudo-product structures. Also we consider the controllability of cone structures and describe the constrained Hamiltonian equations on normal and abnormal geodesics.
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Bibliogr. 32 poz.
  • [1] A. A. Agrachev, Geometry of Optimal Control Problems and Hamiltonian Systems, Nonlinear and Optimal Control Theory, Springer-Verlag, 2004, 1–59.
  • [2] A. A. Agrachev, Rolling balls and octonions, Proceedings Steklov Math. Inst. 258 (2007), 13–22.
  • [3] A. A. Agrachev, Y. L. Sachkov, Control Theory from Geometric Viewpoint, Encyclopaedia of Mathematical Sciences 87, Springer-Verlag, 2004.
  • [4] A. Agrachev, I. Zelenko, Nurowski’s conformal structures for (2, 5)-distributions via dynamics of abnormal extremals, RIMS Kokyuroku 1502 (2006), 204–218.
  • [5] J. C. Baez, J. Huerta, G2 and the rolling ball, Trans. Amer. Math. Soc. 366(10) (2014), 5257–5293.
  • [6] A. Belläiche, The tangent space in sub-Riemannian geometry, Sub-Riemannian Geometry, Progress in Math. 144, Birkhäuser, pp. 1–78, 1996.
  • [7] B. Bonnard, M. Chyba, Singular Trajectories and the Role in Control Theory, Springer-Verlag, 2003.
  • [8] G. Bor, R. Montgomery, G2 and the rolling distributions, Enseign. Math. 55 (2009), 157–196.
  • [9] R. L. Bryant, Élie Cartan and geometric duality, A lecture given at the Institut d’Élie Cartan on 19 June 1998.
  • [10] R. L. Bryant, L. Hsu, Rigidity of integral curves of rank 2 distributions, Invent. Math. 114 (1993), 435–461.
  • [11] E. Cartan, Les systèmes de Pfaff à cinq variables et les équations aux dérivées partielles du second ordre, Ann. Sci. Ecole Norm. Sup. (3) 27 (1910), 109–192.
  • [12] B. Doubrov, I. Zelenko, On local geometry of nonholonomic rank 2 distributions, J. London Math. Soc. (2) 80(3) (2009), 545–566.
  • [13] B. Doubrov, I. Zelenko, Equivalence of variational problems of higher order, Differential Geom. Appl. 29(2) (2011), 255–270.
  • [14] B. Doubrov, I. Zelenko, Prolongation of quasi-principal frame bundles and geometry of flag structures on manifolds, preprint. arXiv math.DG/1210.7334v2
  • [15] G. Ishikawa, Y. Kitagawa, W. Yukuno, Duality of singular paths for (2, 3, 5)-distributions, to appear in Journal of Dynamical and Control Systems, 2014.
  • [16] G. Ishikawa, Y. Machida, Singularities of improper affine spheres and surfaces of constant Gaussian curvature, Internat. J. Math. 17(3) (2006), 269–293.
  • [17] G. Ishikawa, Y. Machida, M. Takahashi, Asymmetry in singularities of tangent surfaces in contact-cone Legendre-null duality, J. Singularities 3 (2011), 126–143.
  • [18] G. Ishikawa, Y. Machida, M. Takahashi, Singularities of tangent surfaces in Cartan’s split G2-geometry, Hokkaido University Preprint Series in Mathematics #1020, 2012, to appear in Asian J. Math.
  • [19] Y. Kitagawa, The Infinitesimal automorphisms of a homogeneous subriemannian contact manifold, Thesis, Nara Women’s University, 2005.
  • [20] W. Liu, H. J. Sussman, Shortest paths for sub-Riemannian metrics on rank-two distributions, Memoirs of Amer. Math. Soc., 118–564, Amer. Math. Soc., 1995.
  • [21] R. Montgomery, A Tour of Subriemannian Geometries, their Geodesics and Applications, Mathematical Surveys and Monographs, 91, Amer. Math. Soc., 2002.
  • [22] T. Morimoto, Cartan connection associated with a subriemannian structure, Differential Geom. Appl. 26(1) (2008), 75–78.
  • [23] P. Nurowski, Differential equations and conformal structures, J. Geom. Phys. 55(1) (2005), 19–49.
  • [24] L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze, E. F. Mishchenko, The Mathematical Theory of Optimal Processes, Pergamon Press, 1964.
  • [25] H. Sato, K. Yamaguchi, Lie tensor product manifolds, Demonstratio Math. 45(4) (2012), 909–927.
  • [26] N. Tanaka, On affine symmetric spaces and the automorphism groups of product manifolds, Hokkaido Math. J. 14 (1980), 277–351.
  • [27] K. Yamaguchi, Differential systems associated with simple graded Lie algebras, Progress in Differential Geometry, Adv. Stud. Pure Math. 22 (1993), 413–494.
  • [28] K. Yamaguchi, T. Yatsui, Parabolic geometries associated with differential equations of finite type, Progress in Math., 252, Birkhäuser, Boston, 2007, pp. 161–209.
  • [29] T. Yatsui, On pseudo-product graded Lie algebras, Hokkaido Math. J. 17 (1988), 333–343.
  • [30] A. M. Vershik, V. Ya. Gershkovich, Nonholonomic dynamical systems, geometry of distributions and variational problems, in Dynamical Systems VII, ed. by V. I. Arnol’d, S. P. Novikov, Encyclopaedia of Math. Sci., 16, Springer-Verlag, 1994, pp. 1–81.
  • [31] I. Zelenko, Fundamental form and the Cartan tensor of (2, 5)-distributions coincide, J. Dynam. Control Systems 12(2) (2006), 247–276.
  • [32] M. Zhitomirskii, Exact normal form for (2, 5) distributions, RIMS Kokyuroku 1502 (2006), 16–28.
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