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 associate 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. 2000 Mathematics Subject Classification. 53A55, 70G45. Differential geometry of rank 2 vector distributions (without additional structures on them) can be treated as well by studying unparameterized curves in Lagrange Grassmannians ([5],[6]) 1 H c = {λ ∈ O : H(λ) = c} is nonempty and consists of regular points of H. Consider the Hamiltonian vector field H on H c , corresponding to the Hamiltonian H, i.e. the vector field satisfying i Hω = −dH, whereω is the canonical symplectic structure on T * M . The integral curves of this Hamiltonian system are normal Pontryagin extremals of the time-optimal problem, associated with geometric structure V, or, shortly, normal extremals of V. For example, if V is a sub-Riemannian structure with underlying distribution D, then the maximized Hamiltonian satisfies H(p, q) = ||p| Dq || q , i.e. H(p, q) is equal to the norm of the restriction of the functional p ∈ T * q M on D q w.r.t. the Euclidean norm || · || q on D q ; O = T * M \D ⊥ , where D ⊥ is the annihilator of D, D ⊥ = {(p, q) ∈ T * M : p(v) = 0 ∀v ∈ D q }. The projections of the trajectories of the corresponding Hamiltonian systems to the base manifold M are normal sub-Riemannian geodesics. If D = T M , then they are exactly the Riemannian geodesics of the corresponding Riemannian structure. Further let H c (q) = H c ∩ T * q M . H c (q) is a codimension 1 submanifold of T * q M . For any λ ∈ H c denote Π λ = T λ H c (π(λ)) , where π : T * M → M is the canonical projection. Actually Π λ is the vertical subspace of T λ H c , (1.2) Π λ = {ξ ∈ T λ H c : π * (ξ) = 0}. Now with ...

In the present paper we construct differential invariants for generic rank 2 vector distributions on n-dimensional manifold. In the case n = 5 (the first case containing functional parameters) E. Cartan found in 1910 the covariant fourth-order tensor invariant for such distributions, using his "reduction-prolongation" procedure (see [12]). After Cartan's work the following questions remained open: first the geometric reason for existence of Cartan's tensor was not clear; secondly it was not clear how to generalize this tensor to other classes of distributions; finally there were no explicit formulas for computation of Cartan's tensor. Our paper is the first in the series of papers, where we develop an alternative approach, which gives the answers to the questions mentioned above. It is based on the investigation of dynamics of the field of so-called abnormal extremals (singular curves) of rank 2 distribution and on the general theory of unparametrized curves in the Lagrange Grassmannian, developed in [4], [5]. In this way we construct the fundamental form and the projective Ricci curvature of rank 2 vector distributions for arbitrary n ≥ 5. For n = 5 we give an explicit method for computation of these invariants and demonstrate it on several examples. In the next paper [19] we show that in the case n = 5 our fundamental form coincides with Cartan's tensor.

Abstract. In 1910 E. Cartan constructed a canonical frame and found the most symmetric case for maximally nonholonomic rank 2 distributions in R 5 . We solve the analogous problem for germs of generic rank 2 distributions in R n for n > 5. We use a completely different approach based on the symplectification of the problem. The main idea is to consider a special odd-dimensional submanifold WD of the cotangent bundle associated with any rank 2 distribution D. It is naturally foliated by characteristic curves, which are also called the abnormal extremals of the distribution D. The dynamics of vertical fibers along characteristic curves defines certain curves of flags of isotropic and coisotropic subspaces in a linear symplectic space. Using the classical theory of curves in projective spaces, we construct the canonical frame of the distribution D on a certain (2n−1)-dimensional fiber bundle over WD with the structure group of all Möbius transformations, preserving 0. The paper is the detailed exposition of the constructions and the results, announced in the short note [8].

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Abstract. We develop an algebraic version of Cartan method of equivalence or an analog of Tanaka prolongation for the (extrinsic) geometry of curves of flags of a vector space W with respect to the action of a subgroup G of the GL(W ). Under some natural assumptions on the subgroup G and on the flags, one can pass from the filtered objects to the corresponding graded objects and describe the construction of canonical bundles of moving frames for these curves in the language of pure Linear Algebra. The scope of applicability of the theory includes geometry of natural classes of curves of flags with respect to reductive linear groups or their parabolic subgroups. As simplest examples, this includes the projective and affine geometry of curves. The case of classical groups is considered in more detail.