02238nas a2200109 4500008004300000245010000043210006900143520184600212100001602058700001802074856003602092 2009 en_Ud 00aJacobi Equations and Comparison Theorems for Corank 1 Sub-Riemannian structures with symmetries0 aJacobi Equations and Comparison Theorems for Corank 1 SubRiemann3 aThe Jacobi curve of an extremal of optimal control problem is a curve in a Lagrangian Grassmannian defined up to a symplectic transformation and containing all information about the solutions of the Jacobi equations along this extremal. In our previous works we constructed the canonical\\nbundle of moving frames and the complete system of symplectic invariants, called curvature maps, for\\nparametrized curves in Lagrange Grassmannians satisfying very general assumptions. The structural\\nequation for a canonical moving frame of the Jacobi curve of an extremal can be interpreted as the\\nnormal form for the Jacobi equation along this extremal and the curvature maps can be seen as the\\n\\\"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 sectional curvature. 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 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 the sub-Riemannian extremals in terms of the bounds for the curvature tensor of this Riemannian manifold and the magnetic field in the case of an 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.1 aChengbo, Li1 aZelenko, Igor uhttp://hdl.handle.net/1963/373600440nas a2200121 4500008004100000245005500041210005400096260003700150300001300187490000700200100001900207856009200226 2002 eng d00aJacobi groups, Jacobi forms and their applications0 aJacobi groups Jacobi forms and their applications aProvidence, RIbAmer. Math. Soc. a99–1110 v311 aBertola, Marco uhttps://www.math.sissa.it/publication/jacobi-groups-jacobi-forms-and-their-applications