Igor Zelenko

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.”

On local geometry of non-holonomic rank 2 distributions

In 1910 E. Cartan constructed a canonical frame and found the most symmetric case for maximally non-holonomic rank 2 distributions in R5. We solve the analogous problem for germs of generic rank 2 distributions in Rn 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 M¨obius transformations, preserving 0. The paper is the detailed exposition of the constructions and the results, announced in the short note (B. Doubrov and I. Zelenko, C. R. Math. Acad. Sci. Paris, Ser. I (8) 342 (2006) 589�594).

On variational approach to differential invariants of rank two distributions

Abstract 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 Cartans 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.

Weakly secure data exchange with Generalized Reed Solomon codes

We focus on secure data exchange among a group of wireless clients. The clients exchange data by broadcasting linear combinations of packets over a lossless channel. The data exchange is performed in the presence of an eavesdropper who has access to the channel and can obtain all transmitted data. Our goal is to develop a weakly secure coding scheme that prevents the eavesdropper from being able to decode any of the original packets held by the clients. We present a randomized algorithm based on Generalized Reed-Solomon (GRS) codes. The algorithm has two key advantages over the previous solutions: it operates over a small (polynomial-size) finite field and provides a way to verify that constructed code is feasible. In contrast, the previous approaches require exponential field size and do not provide an efficient (polynomial-time) algorithm to verify the secrecy properties of the constructed code. We formulate an algebraic-geometric conjecture that implies the correctness of our algorithm and prove its validity for special cases. Our simulation results indicate that the algorithm is efficient in practical settings.

Nonregular Abnormal Extremals of 2-Distribution: Existence, Second Variation, and Rigidity

We study existence and rigidity (W∞1-isolatedness) of nonregular abnormal extremals of completely nonholonomic 2-distribution (nonregularity means that such extremals do not satisfy the strong generalized Legendre–Clebsch condition). Introducing the notion of diagonal form of the second variation, we generalize some results of A. Agrachev and A. Sarychev about rigidity of regular abnormal extremals to the nonregular case. In order to reduce the second variation to the diagonal form, we construct a special curve of Lagrangian subspaces, a Jacobi curve. We show that certain geometric properties of this curve (like simplicity) imply the rigidity of the corresponding abnormal extremal.

On Feedback Classification of Control-Affine Systems with One- and Two-Dimensional Inputs

The paper is devoted to the local classification of generic germs of control-affine systems on an

Fundamental form and the cartan tensor of (2,5)-distributions coincide

n

Geometry of curves in generalized flag varieties

-dimensional manifold with scalar input for any

On Tanaka's Prolongation Procedure for Filtered Structures of Constant Type ?

n\geq4

On geometry of curves of flags of constant type

or with two inputs for