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Boris Doubrov

Publications and source records attributed to Boris Doubrov.

At least 19 recordsLinked to original sources
arXiv2026

$G_2$ and the Maximally Symmetric (3, 8) Distribution with 6-Dimensional Square

In 1910, \'{E}lie Cartan famously realized the split real form of the exceptional Lie group $G_2$ as the symmetry group of the maximally symmetric rank 2 distribution on a 5-dimensional manifold with the small growth vector (2,3,5). In this paper, we discover a new appearance of $G_2$ in the geometric theory of distributions, arising from a rank 3 distribution on an 8-dimensional manifold with the growth vector $(3,6,8)$. The algebra of infinitesimal symmetries of this distribution at any point is 29-dimensional and isomorphic to $(\mathfrak{g}_2 \oplus \mathbb{R}) \ltimes W$, where $\mathfrak{g}_2$ is the Lie algebra of $G_2$ and $W$ is an adjoint module of $\mathfrak{g}_2$. Our model possesses three remarkable properties. First, it is maximally symmetric among all bracket-generating rank 3 distributions with a 6-dimensional square (a family that includes both (3,6,8) and (3,6,7,8) distributions). To the best of our knowledge, this is the first example of a family of distributions defined by a set of prescribed small growth vectors in which maximal symmetry is achieved by a member whose growth vector is not the longest. Second, this model provides the first counterexample to the conjecture that all bracket-generating rank 3 distributions with a 6-dimensional square are of maximal class at a generic point (which is known to hold in dimensions 6 and 7). Third, further analysis yields the control-theoretic consequence that all abnormal extremal trajectories of this model originating at any point of the ambient manifold have a corank of at least 2. To our knowledge, this is the first example with this property among bracket-generating distributions with generic small growth vector for a given rank and ambient dimension. We also give an interpretation of our model in terms of split-octonions, more precisely, in terms of a natural algebraic structure on the tangent bundle to split octonions.

math.DG↗
arXiv2026

Generalized pseudo-product structures and finite type distributions via abnormal extremals

We generalize the classical Tanaka result on the finiteness of the Lie algebra of infinitesimal symmetries for non-degenerate pseudo-product structures to the case when the completely integrable distributions defining the pseudo-product structure are no longer concentrated in the degree $-1$. In order to do this, we modify the notion of universal prolongation of graded nilpotent Lie algebras and generalize the original finiteness criterion of Tanaka. Using this result, we demonstrate that in real analytic category, distributions that are controllable by regular abnormal extremal trajectories, also known as singularly transitive, have finite-dimensional symmetries. This result settles Problem V in the affirmative from the 2013 list of open problems by Andrei Agrachev. Additionally, we discuss applications to symmetries and natural equivalence problems for systems of ODEs of mixed order.

math.DG↗
arXiv2025

Symplectification of Rank 2 Distributions, Normal Cartan Connections, and Cartan Prolongations

We study the Doubrov--Zelenko symplectification procedure for rank $2$ distributions with $5$-dimensional cube -- originally motivated by optimal control theory -- through the lens of Tanaka--Morimoto theory for normal Cartan connections. In this way, for ambient manifolds of dimension $ n \geq 5 $, we prove the existence of the normal Cartan connection associated with the symplectified distribution. Furthermore, we show that this symplectification can be interpreted as the $(n-4)$th iterated Cartan prolongation at a generic point. This interpretation naturally leads to two questions for an arbitrary rank $2$ distribution with $5$-dimensional cube: (1) Is the $(n-4)$th iterated Cartan prolongation the minimal iteration where the Tanaka symbols become unified at generic points? (2) Is the $(n-4)$th iterated Cartan prolongation the minimal iteration admitting a normal Cartan connection via Tanaka--Morimoto theory? Our main results demonstrate that: (a) For $n > 5$, the answer to the second question is positive (in contrast to the classical $n = 5$ case from $G_2$-parabolic geometries); (b) For $n \geq 5$, the answer to the first question is negative: unification occurs already at the $(n-5)$th iterated Cartan prolongation.

math.DG↗
arXiv2023

Extrinsic geometry and linear differential equations of $\mathfrak{sl}_3$-type

As an application of the general theory on extrinsic geometry, we investigate extrinsic geometry in frag varieties and systems of linear PDE's for a class of special interest associated with the adjoint representation of $\mathfrak{sl}(3)$. We carry out a complete local classification of the homogeneous structures in this class. As a result, we find 7 kinds of new systems of linear PDE's of second order on a 3-dimensional contact manifold each of which has a solution space of dimension 8. Among them there are included a system of PDE's called contact Cayley's surface and one which has $\mathfrak{sl}(2)$ symmetry.

math.DG↗
arXiv2021

Extrinsic Geometry and Linear Differential Equations

We give a unified method for the general equivalence problem of extrinsic geometry, on the basis of our formulation of a general extrinsic geometry as that of an osculating map $φ\colon (M,\mathfrak f) \to L/L^0 \subset \operatorname{Flag}(V,ϕ)$ from a filtered manifold $(M,\mathfrak f)$ to a homogeneous space $L/L^0$ in a flag variety $\operatorname{Flag}(V,ϕ)$, where $L$ is a finite-dimensional Lie group and $L^0$ its closed subgroup. We establish an algorithm to obtain the complete systems of invariants for the osculating maps which satisfy the reasonable regularity condition of constant symbol of type $(\mathfrak g_-, \operatorname{gr} V, L)$. We show the categorical isomorphism between the extrinsic geometries in flag varieties and the (weighted) involutive systems of linear differential equations of finite type. Therefore we also obtain a complete system of invariants for a general involutive systems of linear differential equations of finite type and of constant symbol. The invariants of an osculating map (or an involutive system of linear differential equations) are proved to be controlled by the cohomology group $H^1_+(\mathfrak g_-, \mathfrak l / \bar{\mathfrak g})$ which is defined algebraically from the symbol of the osculating map (resp. involutive system), and which, in many cases (in particular, if the symbol is associated with a simple Lie algebra and its irreducible representation), can be computed by the algebraic harmonic theory, and the vanishing of which gives rigidity theorems in various concrete geometries. We also extend the theory to the case when $L$ is infinite dimensional.

math.DG↗
arXiv2021

Classification of simply-transitive Levi non-degenerate hypersurfaces in $\mathbb{C}^3$

Holomorphically homogeneous CR real hypersurfaces $M^3 \subset \mathbb{C}^2$ were classified by Élie Cartan in 1932. In the next dimension, we complete the classification of simply-transitive Levi non-degenerate hypersurfaces $M^5 \subset \mathbb{C}^3$ using a novel Lie algebraic approach independent of any earlier classifications of abstract Lie algebras. Central to our approach is a new coordinate-free formula for the fundamental (complexified) quartic tensor. Our final result has a unique (Levi-indefinite) non-tubular model, for which we demonstrate geometric relations to planar equi-affine geometry.

math.DG↗
arXiv2020

Vector distributions with very large symmetries via rational normal curves

We construct a sequence of rank 3 distributions on $n$-dimensional manifolds for any $n\geq 7$ such that the dimension of their symmetry group grows exponentially in $n$ (more precisely it is equal to $\operatorname{Fib}_{n-1}+n+2$, where $\operatorname{Fib}_n$ is the $n$-th Fibonacci number, starting with $\operatorname{Fib}_1=\operatorname{Fib}_2=1$) and such that the maximal order of weighted jet needed to determine these symmetries grows quadratically in $n$. These examples are in sharp contrast with the parabolic geometries where the dimension of a symmetry group grows polynomially with respect to the dimension of the ambient manifold and the corresponding maximal order of weighted jet space is equal to the degree of nonholonomy of the underlying distribution plus $1$. Our models are closely related to the geometry of certain curves of symplectic flags and of the rational normal curves.

math.DG↗
arXiv2017

On C-class equations

The concept of a C-class of differential equations goes back to E. Cartan with the upshot that generic equations in a C-class can be solved without integration. While Cartan's definition was in terms of differential invariants being first integrals, all results exhibiting C-classes that we are aware of are based on the fact that a canonical Cartan geometry associated to the equations in the class descends to the space of solutions. For sufficiently low orders, these geometries belong to the class of parabolic geometries and the results follow from the general characterization of geometries descending to a twistor space. In this article we answer the question of whether a canonical Cartan geometry descends to the space of solutions in the case of scalar ODEs of order at least four and of systems of ODEs of order at least three. As in the lower order cases, this is characterized by the vanishing of the generalized Wilczynski invariants, which are defined via the linearization at a solution. The canonical Cartan geometries (which are not parabolic geometries) are a slight variation of those available in the literature based on a recent general construction. All the verifications needed to apply this construction for the classes of ODEs we study are carried out in the article, which thus also provides a complete alternative proof for the existence of canonical Cartan connections associated to higher order (systems of) ODEs.

math.DG↗
arXiv2017

Integrable systems in 4D associated with sixfolds in Gr(4,6)

Let $Gr(d,n)$ be the Grassmannian of $d$-dimensional linear subspaces of an $n$-dimensional vector space $V$. A submanifold $X\subset Gr(d, n)$ gives rise to a differential system $Σ(X)$ that governs $d$-dimensional submanifolds of $V$ whose Gaussian image is contained in $X$. We investigate a special case of this construction where $X$ is a sixfold in $Gr(4, 6)$. The corresponding system $Σ(X)$ reduces to a pair of first-order PDEs for 2 functions of 4 independent variables. Equations of this type arise in self-dual Ricci-flat geometry. Our main result is a complete description of integrable systems $Σ(X)$. These naturally fall into two subclasses. (1) Systems of Monge-Ampère type. The corresponding sixfolds $X$ are codimension 2 linear sections of the Plücker embedding $Gr(4,6)\subset\mathbb{P}^{14}$. (2) General linearly degenerate systems. The corresponding sixfolds $X$ are the images of quadratic maps $\mathbb{P}^6- Gr(4, 6)$ given by a version of the classical construction of Chasles. We prove that integrability is equivalent to the requirement that the characteristic variety of system $Σ(X)$ gives rise to a conformal structure which is self-dual on every solution. In fact, all solutions carry hyper-Hermitian geometry.

nlin.SI↗
arXiv2017

Three-dimensional homogeneous spaces with non-solvable transformation groups

We classify all transitive actions of Lie algebras of vector fields on C^3 and R^3 up to a local equivalence and discuss why this classification can not be extended in general to the solvable case. The main technical tool is the structure of one-dimensional invariant foliations on homogeneous spaces.

math.DG↗
arXiv2017

On a class of integrable systems of Monge-Ampère type

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

nlin.SI↗
arXiv2016

On the integrability in Grassmann geometries: integrable systems associated with fourfolds Gr(3, 5)

We investigate dispersionless integrable systems in 3D associated with fourfolds in the Grassmannian Gr(3,5). Such systems appear in numerous applications in continuum mechanics, general relativity and differential geometry, and include such well-known examples as the dispersionless Kadomtsev-Petviashvili equation, the Boyer-Finley equation, etc. We prove the equivalence of the four different approaches to integrability, revealing a remarkable correspondence with Einstein-Weyl geometry and the theory of GL(2,R) structures.

math.DG↗
arXiv2016

On local geometry of vector distributions with given Jacobi symbols

Based on the ideas of Optimal Control, we introduce the new basic characteristic of a bracket generating distribution, the Jacobi symbol. In contrast to the classical Tanaka symbol, the set of Jacobi symbols is discrete and classifiable. We give an explicit and unified algebraic procedure for the construction of the canonical frames (the structure of absolute parallelism) for all distribution with given Jacobi symbol. We describe all Jacobi symbols for which this procedure ends up in a finite number of steps (i.e. all Jacobi symbols of finite type) and distinguish the symplectically flat distributions with given Jacobi symbol, which in the case of finite type are the maximally symmetric distribution among all distribution with given Jacobi symbol. Also, for the most of Jacobi symbols we relate the prolongation procedure and the resulting prolongation algebra, which is the algebra of infinitesimal symmetries of the corresponding symplectically flat distribution, to the standard (in the sense of Kobayashi and Sternberg) prolongation of certain subspace in the algebra of infinitesimal symmetries of the flat (unparametrized) curve of flags associated with the Jacobi symbol. Finally, we give an upper bounds for the size of the resulting prolongation algebra in terms of certain spaces of polynomials vanishing on certain projective varieties. For some class of Jacobi symbols we are able to identify the resulting prolongation algebra with such spaces of polynomials. In this way we get the description of the prolongation algebra for all Jacobi symbols of finite type appearing in rank 2 and rank 3 distributions and also for most of such Jacobi symbols appearing in rank 4 distributions. In particular, in the case of rank 3 distributions this description is given in terms of the tangential developable and the secant varieties of a rational normal curve.

math.DG↗
arXiv2016

Homogeneous integrable Legendrian contact structures in dimension five

We consider Legendrian contact structures on odd-dimensional complex analytic manifolds. We are particularly interested in integrable structures, which can be encoded by compatible complete systems of second order PDEs on a scalar function of many independent variables and considered up to point transformations. Using the techniques of parabolic differential geometry, we compute the associated regular, normal Cartan connection and give explicit formulas for the harmonic part of the curvature. The PDE system is trivializable by means of point transformations if and only if the harmonic curvature vanishes identically. In dimension five, the harmonic curvature takes the form of a binary quartic field, so there is a Petrov classification based on its root type. We give a complete local classification of all five-dimensional integrable Legendrian contact structures whose symmetry algebra is transitive on the manifold and has at least one-dimensional isotropy algebra at any point.

math.DG↗
arXiv2016

The geometry of second-order ordinary differential equations

These are lecture notes of the Summer school on the geometry of differential equations held in Nordfjordeid, Norway in 1996. They cover geometric structures related to scalar second order ODEs, the construction of the associated Cartan connection, techniques for computing invariants of differential equations starting from the Cartan connection, recognition of symmetric 2nd order ODEs by their differential invariants and the constructive aspects of the local equivalence problem.

math.DG↗
arXiv2014

Fundamental invariants of systems of ODEs of higher order

We find the complete set of fundamental invariants for systems of ordinary differential equations of order $\ge 4$ under the group of point transformations generalizing similar results for contact invariants of a single ODE and point invariants of systems of the second and the third order. It turns out that starting from systems of order $(k+1)\ge 4$, the complete set of fundamental invariants is formed by $k$ generalized Wilczynski invariants coming from the linearized system and an additional invariant of degree 2.

math.DG↗