Jacobi equations and comparison theorems for corank 1 sub-Riemannian structures with symmetries.

*(English)*Zbl 1216.53039Summary: The Jacobi curve of an extremal of an optimal control problem is a curve in a Lagrangian Grassmannian defined up to a symplectic transformation containing all information about the solutions of the Jacobi equations along this extremal. In our previous works, we constructed the canonical bundle of moving frames and the complete system of symplectic invariants, called curvature maps, for parametrized curves in Lagrange Grassmannians satisfying very general assumptions. The structural equation for a canonical moving frame of the Jacobi curve of an extremal can be interpreted as the normal form for the Jacobi equation along this extremal and the curvature maps can be seen as the “coefficients” of this normal form. In the case of a Riemannian metric, there is only one curvature map and it is naturally related to the Riemannian curvature tensor.

In the present paper, we study the curvature maps for a sub-Riemannian structure on a corank 1 distribution having an additional transversal infinitesimal symmetry. After the factorization by the integral foliation of this symmetry, such a sub-Riemannian structure can be reduced to a Riemannian manifold equipped with a closed 2-form (a magnetic field). We obtain explicit expressions for the curvature maps of the original sub-Riemannian structure in terms of the curvature tensor of this Riemannian manifold and the magnetic field. We also estimate the number of conjugate points along sub-Riemannian extremals in terms of the bounds for the curvature tensor of this Riemannian manifold and the magnetic field in the case of a uniform magnetic field. The language developed for the calculation of the curvature maps can be applied to more general sub-Riemannian structures with symmetries, including sub-Riemmannian structures appearing naturally in Yang-Mills fields.

In the present paper, we study the curvature maps for a sub-Riemannian structure on a corank 1 distribution having an additional transversal infinitesimal symmetry. After the factorization by the integral foliation of this symmetry, such a sub-Riemannian structure can be reduced to a Riemannian manifold equipped with a closed 2-form (a magnetic field). We obtain explicit expressions for the curvature maps of the original sub-Riemannian structure in terms of the curvature tensor of this Riemannian manifold and the magnetic field. We also estimate the number of conjugate points along sub-Riemannian extremals in terms of the bounds for the curvature tensor of this Riemannian manifold and the magnetic field in the case of a uniform magnetic field. The language developed for the calculation of the curvature maps can be applied to more general sub-Riemannian structures with symmetries, including sub-Riemmannian structures appearing naturally in Yang-Mills fields.

##### MSC:

53C17 | Sub-Riemannian geometry |

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

49J15 | Existence theories for optimal control problems involving ordinary differential equations |

34C10 | Oscillation theory, zeros, disconjugacy and comparison theory for ordinary differential equations |

##### Keywords:

sub-Riemannian structures; Jacobi equations; curves in Lagrange Grassmannians; conjugate points; comparison theorems; magnetic field on Riemannian manifolds
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\textit{C. Li} and \textit{I. Zelenko}, J. Geom. Phys. 61, No. 4, 781--807 (2011; Zbl 1216.53039)

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##### References:

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