Differential geometry of curves in Lagrange Grassmannians with given Young diagram.

*(English)*Zbl 1177.53020Summary: Curves in Lagrange Grassmannians appear naturally in the intrinsic study of geometric structures on manifolds. By a smooth geometric structure on a manifold we mean any submanifold of its tangent bundle, transversal to the fibers. One can consider the time-optimal problem naturally associated with a geometric structure. The Pontryagin extremals of this optimal problem are integral curves of certain Hamiltonian system in the cotangent bundle. The dynamics of the fibers of the cotangent bundle w.r.t. this system along an extremal is described by certain curve in a Lagrange Grassmannian, called Jacobi curve of the extremal. Any symplectic invariant of the Jacobi curves produces the invariant of the original geometric structure. The basic characteristic of a curve in a Lagrange Grassmannian is its Young diagram. The number of boxes in its \(k\)-th column is equal to the rank of the \(k\)th derivative of the curve (which is an appropriately defined linear mapping) at a generic point. We will describe the construction of the complete system of symplectic invariants for parameterized curves in a Lagrange Grassmannian with given Young diagram. It allows to develop in a unified way local differential geometry of very wide classes of geometric structures on manifolds, including both classical geometric structures such as Riemannian and Finslerian structures and less classical ones such as sub-Riemannian and sub-Finslerian structures, defined on nonholonomic distributions.

##### MSC:

53B25 | Local submanifolds |

53D12 | Lagrangian submanifolds; Maslov index |

16G20 | Representations of quivers and partially ordered sets |

70G45 | Differential geometric methods (tensors, connections, symplectic, Poisson, contact, Riemannian, nonholonomic, etc.) for problems in mechanics |

##### Keywords:

curves in Lagrange Grassmannians; moving frames; Young diagrams; sub-Riemannian structures; symplectic invariants; quivers
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\textit{I. Zelenko} and \textit{C. Li}, Differ. Geom. Appl. 27, No. 6, 723--742 (2009; Zbl 1177.53020)

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##### References:

[1] | Agrachev, A., Any sub-Riemannian metric has points of smoothness, Russian math. dokl.79, 1-3, (2009) · Zbl 1253.53029 |

[2] | Agrachev, A.A.; Gamkrelidze, R.V., Feedback-invariant optimal control theory. I. regular extremals, J. dynam. control systems, 3, 3, 343-389, (1997) · Zbl 0952.49019 |

[3] | Agrachev, A.; Zelenko, I., Geometry of Jacobi curves. I, J. dynam. control systems, 8, 1, 93-140, (2002) · Zbl 1019.53038 |

[4] | Agrachev, A.; Zelenko, I., Geometry of Jacobi curves. II, J. dynam. control systems, 8, 2, 167-215, (2002) · Zbl 1045.53051 |

[5] | Derksen, H.; Weyman, J., Quiver representations, Notices amer. math. soc., 52, 2, 200-206, (2005) · Zbl 1143.16300 |

[6] | Doubrov, B.; Zelenko, I., A canonical frame for nonholonomic rank two distributions of maximal class, C. R. acad. sci. Paris, ser. I, 342, 8, 589-594, (2006) · Zbl 1097.58002 |

[7] | B. Doubrov, I. Zelenko, On local geometry of nonholonomic rank 2 distributions, J. London Math. Soc., in press, arXiv: math.DG/0703662, 21 p · Zbl 1202.58002 |

[8] | C. Li, I. Zelenko, Jacobi equations and comparison theorems for corank 1 sub-Riemannian structures with symmetries, in preparation · Zbl 1216.53039 |

[9] | Ovsienko, V., Lagrange Schwarzian derivative and symplectic Sturm theory, Annales de la faculté des sciences de Toulouse Sér. 6, 2, 1, 73-96, (1993) · Zbl 0780.34004 |

[10] | Pontryagin, L.S.; Boltyanskii, V.G.; Gamkrelidze, R.V.; Mischenko, E.F., The mathematical theory of optimal processes, (1962), Wiley New York · Zbl 0102.32001 |

[11] | I. Zelenko, Complete systems of invariants for rank 1 curves in Lagrange Grassmannians, Differential Geometry and Its Applications, Proc. Conf. Prague, August 30-September 3, 2004, Charles University, Prague, 2005, pp. 365-379 |

[12] | Zelenko, I.; Li, C., Parametrized curves in Lagrange Grassmannians, C. R. acad. sci. Paris, ser. I, 345, 11, 647-652, (2007) · Zbl 1130.53042 |

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