Boris Doubrov

On the integrability of symplectic Monge–Ampère equations

Abstract Let u be a function of n independent variables x 1 , … , x n , and let U = ( u i j ) be the Hessian matrix of u . The symplectic Monge–Ampere equation is defined as a linear relation among all possible minors of U . Particular examples include the equation det U = 1 governing improper affine spheres and the so-called heavenly equation, u 13 u 24 − u 23 u 14 = 1 , describing self-dual Ricci-flat 4 -manifolds. In this paper we classify integrable symplectic Monge–Ampere equations in four dimensions (for n = 3 the integrability of such equations is known to be equivalent to their linearisability). This problem can be reformulated geometrically as the classification of ‘maximally singular’ hyperplane sections of the Plucker embedding of the Lagrangian Grassmannian. We formulate a conjecture that any integrable equation of the form F ( u i j ) = 0 in more than three dimensions is necessarily of the symplectic Monge–Ampere type.

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).

Generalized Wilczynski Invariants for Non-Linear Ordinary Differential Equations

We show that classical Wilczynski-Se-ashi invariants of linear systems of ordinary differential equations are generalized in a natural way to contact invariants of non-linear ODEs. We explore geometric structures associated with equations that have vanishing generalized Wilczynski invariants and establish relationship of such equations with deformation theory of rational curves on complex algebraic surfaces.

Geometry of curves in generalized flag varieties

The current paper is devoted to the study of integral curves of constant type in generalized flag varieties. We construct a canonical moving frame bundle for such curves and give a criterion when it turns out to be a Cartan connection. Generalizations to parametrized curves, to higher-dimensional submanifolds and to parabolic geometries are discussed.

On geometry of curves of flags of constant type

We develop an algebraic version of Cartan’s 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 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.

Contact Lie algebras of vector fields on the plane.

The paper is devoted to the complete classication of all real Lie algebras of contact vector elds on the rst jet space of one-dimensional submanifolds in the plane. This completes Sophus Lies classication of all possible Lie algebras of contact symmetries for ordinary dierential equations. As a main tool we use the abstract theory of ltered and graded Lie algebras. We also describe all dierential and integral invariants of new Lie algebras found in the paper and discuss the innite-dimensional case.

Co-calibrated G2 structure from cuspidal cubics

We establish a twistor correspondence between a cuspidal cubic curve in a complex projective plane, and a co-calibrated homogeneous G2 structure on the seven-dimensional parameter space of such cubics. Imposing the Riemannian reality conditions leads to an explicit co-calibrated G2 structure on SU(2, 1)/U(1). This is an example of an SO(3) structure in seven dimensions. Cuspidal cubics and their higher degree analogues with constant projective curvature are characterised as integral curves of certain seventh order ODEs. Projective orbits of such curves are shown to be analytic continuations of Aloff–Wallach manifolds, and it is shown that only cubics lift to a complete family of contact rational curves in a projectivised cotangent bundle to a projective plane.

On a class of integrable systems of Monge-Ampère type

We investigate a class of multi-dimensional two-component systems of Monge-Ampere type that can be viewed as generalisations of heavenly type equations appearing in a self-dual Ricci-flat geometry. Based on the Jordan-Kronecker theory of the skew-symmetric matrix pencils, a classification of normal forms of such systems is obtained. All two-component systems of Monge-Ampere type turn out to be integrable and can be represented as the commutativity conditions of parameter-dependent vector fields. Geometrically, systems of Monge-Ampere type are associated with linear sections of the Grassmannians. This leads to an invariant differential-geometric characterisation of the Monge-Ampere property.

On geometry of affine control systems with one input

We demonstrate how the novel approach to the local geometry of structures of nonholonomic nature, originated by Andrei Agrachev, works for rank 2 distributions of maximal class in ℝn with additional structures such as affine control systems with one input spanning these distributions, sub-(pseudo)Riemannian structures etc. In contrast to the case of an arbitrary rank 2 distribution without additional structures, in the considered cases each abnormal extremal (of the underlying rank 2 distribution) possesses a distinguished parametrization. This fact allows one to construct the canonical frame on a (2n−3)-dimensional for arbitrary n ≥ 5. The moduli spaces of the most symmetric models are described as well.

Geometry of rank 2 distributions with nonzero Wilczynski invariants

In the famous 1910 “cinq variables” paper Cartan showed in particular that for maximally nonholonomic rank 2 distributions in ℝ5 with non-zero covariant binary biquadratic form the dimension of the pseudo-group of local symmetries does not exceed 7 and among such distributions he described the one-parametric family of distributions for which this pseudo-group is exactly 7-dimensional. Using the novel interpretation of the Cartan covariant binary biquadratic form via the classical Wilczynski invariant of curves in projective spaces associated with abnormal extremals of the distributions [4, 27, 28] one can generalize this Cartan result to rank 2 distributions in ℝn satisfying certain genericity assumption, called maximality of class, for arbitrary n ≥ 5. In the present paper for any rank 2 distribution of maximal class with at least one nonvanishing generalized Wilczynski invariants we construct the canonical frame on a (2n — 3)-dimensional bundle and describe explicitly the moduli spaces of the most symmetric models. The relation of our results to the divergence equivalence of Lagrangians of higher order is given as well.