Lagrange Grassmannian Research Articles

Jacobi equations and Comparison Theorems for corank 1 sub-Riemannian structures with symmetries

The 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 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 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 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.

Differential geometry of curves in Lagrange Grassmannians with given Young diagram

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 kth column is equal to the rank of the kth 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.

Parametrized curves in Lagrange Grassmannians

Curves in Lagrange Grassmannians naturally appear when one studies Jacobi equations for extremals, associated with geometric structures on manifolds. We fix integers d i and consider curves Λ ( t ) for which at each t the derivatives of order ⩽ i of all curves of vectors ℓ ( t ) ∈ Λ ( t ) span a subspace of dimension d i . We will describe the construction of a complete system of symplectic invariants for such parametrized curves, satisfying a certain genericity assumption, and give applications to geometric structures, including sub-Riemannian and sub-Finslerian structures. To cite this article: I. Zelenko, C. Li, C. R. Acad. Sci. Paris, Ser. I 345 (2007).

On variational approach to differential invariants of rank two distributions

We construct differential invariants for generic rank 2 vector distributions on n-dimensional manifolds, where n ⩾ 5 . Our method for the construction of invariants is completely different from the Cartan reduction-prolongation procedure. It is based on the dynamics of the field of so-called abnormal extremals (singular curves) of rank 2 distribution and on the theory of unparameterized curves in the Lagrange Grassmannian, developed in [A. Agrachev, I. Zelenko, Geometry of Jacobi curves I, J. Dynam. Control Syst. 8 (1) (2002) 93–140; II, 8 (2) (2002) 167–215]. In this way we construct the fundamental form and the projective Ricci curvature of rank 2 vector distributions for arbitrary n ⩾ 5 . In the next paper [I. Zelenko, Fundamental form and Cartan's tensor of (2,5)-distributions coincide, J. Dynam. Control. Syst., in press, SISSA preprint, Ref. 13/2004/M, February 2004, math.DG/0402195] we show that in the case n = 5 our fundamental form coincides with the Cartan covariant biquadratic binary form, constructed in 1910 in [E. Cartan, Les systemes de Pfaff a cinque variables et les equations aux derivees partielles du second ordre, Ann. Sci. Ecole Normale 27 (3) (1910) 109–192; reprinted in: Oeuvres completes, Partie II, vol. 2, Gautier-Villars, Paris, 1953, pp. 927–1010]. Therefore first our approach gives a new geometric explanation for the existence of the Cartan form in terms of an invariant degree four differential on an unparameterized curve in Lagrange Grassmannians. Secondly, our fundamental form provides a natural generalization of the Cartan form to the cases n > 5 . Somewhat surprisingly, this generalization yields a rational function on the fibers of the appropriate vector bundle, as opposed to the polynomial function occurring when n = 5 . For n = 5 we give an explicit method for computing our invariants and demonstrate the method on several examples.

Geometry of Jacobi Curves. I

Jacobi curves are deep generalizations of the spaces of “Jacobi fields” along Riemannian geodesics. Actually, Jacobi curves are curves in the Lagrange Grassmannians. In our paper we develop differential geometry of these curves which provides basic feedback or gauge invariants for a wide class of smooth control systems and geometric structures. Two principal invariants are the generalized Ricci curvature, which is an invariant of the parametrized curve in the Lagrange Grassmannian endowing the curve with a natural projective structure, and a fundamental form, which is a fourth-order differential on the curve. The so-called rank 1 curves are studied in more detail. Jacobi curves of this class are associated with systems with scalar controls and with rank 2 vector distributions. In the forthcoming second part of the paper we will present the comparison theorems (i.e., the estimates for the conjugate points in terms of our invariants( for rank 1 curves an introduce an important class of “flat curves.”

Hermitian symplectic geometry and extension theory

Here we give brief account of Hermitian symplectic spaces, showing that they are intimately connected to symmetric as well as self-adjoint extensions of a symmetric operator. Furthermore, we find an explicit parametrization of the Lagrange Grassmannian in terms of the unitary matrices . This allows us to explicitly describe all self-adjoint boundary conditions for the Schrodinger operator on the graph in terms of a unitary matrix. We show that the asymptotics of the scattering matrix can be expressed simply in terms of this unitary matrix.

Quadratic mappings in geometric control theory

The article is dedicated to local investigation of mappings of type “entry-exit” of smooth controlled systems. The homological theory of quadratic mappings and the geometry of the Lagrange Grassmannian are used for the study of sets of the level and the form of mappings of type “entry-exit,” including for the obtaining of the necessary and sufficient conditions for local optimality.

HOLOMORPHIC EXTENSIONS OF REPRESENTATIONS OF THE GROUP OF DIFFEOMORPHISMS OF THE CIRCLE

This paper gives the construction of a semigroup ? which could be thought of as the complexincation of the group Diff of analytic diffeomorphisms of the circle, and it is shown that any unitary projective representation of Diff with highest weight has a holomorphic extension to ?: For this, ? is embedded in the semigroup of endomorphisms of canonical commutation relations (this is a certain part of the Lagrange Grassmannian in complex symplectic Hilbert space). Bibliography: 25 titles.