arXiv subjects

Boris Doubrov

Publications and source records attributed to Boris Doubrov.

34 records · Page 2Linked to original sources

arXiv2013

Prolongation of quasi-principal frame bundles and geometry of flag structures on manifolds

Motivated by the geometric theory of differential equations and the variational approach to the equivalence problem for geometric structures on manifolds, we consider the problem of equivalence for distributions with fixed submanifolds of flags on each fiber. We call them flag structures. The construction of the canonical frames for these structures can be given in the two prolongation steps: the first step, based on our previous works, gives the canonical bundle of moving frames for the fixed submanifolds of flags on each fiber and the second step consists of the prolongation of the bundle obtained in the first step. The bundle obtained in the first step is not as a rule a principal bundle so that the classical Tanaka prolongation procedure for filtered structures can not be applied to it. However, under natural assumptions on submanifolds of flags and on the ambient distribution, this bundle satisfies a nice weaker property. The main goal of the present paper is to formalize this property, introducing the so-called quasi-principle frame bundles, and to generalize the Tanaka prolongation procedure to these bundles. Applications to the equivalence problems for systems of differential equations of mixed order, bracket generating distributions, sub-Riemannian and more general structures on distributions are given.

math.DG↗

arXiv2013

Symmetries of trivial systems of ODEs of mixed order

We compute symmetry algebras of a system of two equations y^(k)=z^(l)=0, where 2<=k<l. It appears that there are many ways to convert such system of ODEs to an exterior differential system. They lead to different series of finite-dimensional symmetry algebras. For example, for (k,l)=(2,3) we get two non-isomorphic symmetry algebras of the same dimension. We explore how these symmetry algebras are related to both Sternberg prolongation of G-structures and Tanaka prolongation of graded nilpotent Lie algebras. Surprisingly, the case (k,l)=(2,3) provides an example of a linear subalgebra g in gl(5,R) such that the Sternberg prolongations of g and g^t are both of the same dimension, but are non-isomorphic. We also discuss the non-linear case and the link with flag structures on smooth manifolds.

math.DG↗

arXiv2013

A new example of a generic 2-distribution on a 5-manifold with large symmetry algebra

We discover a new example of a generic rank 2-distribution on a 5-manifold with a 6-dimensional transitive symmetry algebra, which is not present in Cartan's classical five variables paper. It corresponds to the Monge equation z' = y + (y'')^(1/3) with invariant quartic having root type [4], and a 6-dimensional non-solvable symmetry algebra isomorphic to the semidirect product of sl(2) and the 3-dimensional Heisenberg algebra.

math.DG↗

arXiv2013

Maximally degenerate Weyl tensors in Riemannian and Lorentzian signatures

We establish the submaximal symmetry dimension for Riemannian and Lorentzian conformal structures. The proof is based on enumerating all subalgebras of orthogonal Lie algebras of sufficiently large dimension and verifying if they stabilize a non-zero Weyl tensor up to scale. Our main technical tools include Dynkin's classification of maximal subalgebras in complex simple Lie algebras, a theorem of Mostow, and Kostant's Bott-Borel-Weil theorem.

math.DG↗

arXiv2013

Geometry of rank 2 distributions with nonzero Wilczynski invariants and 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 in the following two situations: rank 2 distributions of maximal class in R^n with non-zero generalized Wilczynski invariants and rank 2 distributions of maximal class in R^n with additional structures such as affine control system with one input spanning these distributions, sub-(pseudo)Riemannian structures etc. The common feature of these two situations is that 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 bundle in both situations for arbitrary n greater than 4. The moduli spaces of the most symmetric models for both situations are described as well. The relation of our results to the divergence equivalence of Lagrangians of higher order is given

math.DG↗

arXiv2013

Geometry of curves in parabolic homogeneous spaces

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

math.DG↗

arXiv2012

Co--calibrated $G_2$ structure from cuspidal cubics

We establish a twistor correspondence between a cuspidal cubic curve in a complex projective plane, and a co-calibrated homogeneous $G_2$ structure on the seven--dimensional parameter space of such cubics. Imposing the Riemannian reality conditions leads to an explicit co-calibrated $G_2$ 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 7th 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.

math.DG↗

arXiv2011

On geometry of curves of flags of constant type

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.

math.DG↗

arXiv2010

Equivalence of variational problems of higher order

We show that for n>2 the following equivalence problems are essentially the same: the equivalence problem for Lagrangians of order n with one dependent and one independent variable considered up to a contact transformation, a multiplication by a nonzero constant, and modulo divergence; the equivalence problem for the special class of rank 2 distributions associated with underdetermined ODEs z'=f(x,y,y',..., y^{(n)}); the equivalence problem for variational ODEs of order 2n. This leads to new results such as the fundamental system of invariants for all these problems and the explicit description of the maximally symmetric models. The central role in all three equivalence problems is played by the geometry of self-dual curves in the projective space of odd dimension up to projective transformations via the linearization procedure (along the solutions of ODE or abnormal extremals of distributions). More precisely, we show that an object from one of the three equivalence problem is maximally symmetric if and only if all curves in projective spaces obtained by the linearization procedure are rational normal curves.

math.DG↗

arXiv2010

Graded nilpotent Lie algebras of infinite type

The paper gives the complete characterization of all graded nilpotent Lie algebras with infinite-dimensional Tanaka prolongation as extensions of graded nilpotent Lie algebras of lower dimension by means of a commutative ideal. We introduce a notion of weak characteristics of a vector distribution and prove that if a bracket-generating distribution of constant type does not have non-zero complex weak characteristics, then its symmetry algebra is necessarily finite-dimensional. The paper also contains a number of illustrative algebraic and geometric examples including the proof that any metabelian Lie algebra with a 2-dimensional center always has an infinite-dimensional Tanaka prolongation.

math.DG↗

arXiv2009

Space-time and G_2

A Weyl structure is a bundle over space-time, whose fiber at each space-time point is a space of maximally isotropic complex tangent planes. We develop the theory of Weyl connections for Weyl structures and show that the requirement that the connection be torsion-free fixes the Weyl connection uniquely. Further we show that to each such Weyl connection, there is naturally associated a (2, 3, 5)-Pfaffian system, as first analyzed by Cartan. We determine the associated G_2-conformal structure and calculate it explicitly in the cases of the Kapadia family of space-times and of the Schwarzschild solution

gr-qc↗

arXiv2008

Inclusions between parabolic geometries

Some of the well known Fefferman like constructions of parabolic geometries end up with a new structure on the same manifold. In this paper, we classify all such cases with the help of the classical Onishchik's lists \cite{onish1} and we treat in detail the only new series of inclusions providing the spinorial structures on the manifolds with generic free distributions. Our technique relies on the cohomological understanding of the canonical normal Cartan connections for parabolic geometries and the classical computations with exterior forms. Apart of the complete discussion of the distributions from the geometrical point of view and the new functorial construction of the inclusion into the spinorial geometry, we also discuss the normality problem of the resulting spinorial connections. In particular, there is a non--trivial subclass of distributions providing normal spinorial connections directly by the construction.

math.DG↗

arXiv2008

On local geometry of rank 3 distributions with 6-dimensional square

We solve the equivalence problem for rank 3 completely nonholonomic vector distributions with 6-dimensional square on a smooth manifold of arbitrary dimension n under very mild genericity conditions. The main idea is to consider the projectivization of the annihilator of a given 3-dimensional distribution. It is naturally foliated by characteristic curves, which are also called the abnormal extremals of the distribution. The dynamics of vertical fibers along characteristic curves defines certain curves of flags of isotropic and coisotropic subspaces in a linear symplectic space. The problem of equivalence of distributions can be essentially reduced to the differential geometry of such curves. The class of all 3-distributions under consideration is split into a finite number of subclasses according to the Young diagram of their flags. The local geometry of distributions can be recovered from the properties of the symmetry group of so-called flat curves of flags associated with this Young diagram. In each subclass we describe the flat distribution and construct a canonical frame for any other distribution. It turns out that for n>6 in the most nontrivial case the symmetry algebra of the flat distribution can be described in terms of rational normal curves (their secants and tangential developables) in projective spaces and its dimension grows exponentially with respect to n.

math.DG↗

arXiv2007

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.

math.DG↗

arXiv2007

On local geometry of nonholonomic rank 2 distributions

In 1910 E. Cartan constructed a canonical frame and found the most symmetric case for maximally nonholonomic rank 2 distributions in $\mathbb R^5$. We solve the analogous problem for germs of generic rank 2 distributions in ${\mathbb R}^n$ 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 $W_D$ 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 $W_D$ with the structure group of all Möbius transformations, preserving 0.

math.DG↗

arXiv2005

A Canonical Frame for Nonholonomic Rank Two Distributions of Maximal Class

In 1910 E. Cartan constructed the canonical frame and found the most symmetric case for maximally nonholonomic rank 2 distributions on a 5-dimensional manifold. We solve the analogous problems for rank 2 distributions on an n-dimensional manifold for arbitrary n greater than 5. Our method is a kind of symplectification of the problem and it is completely different from the Cartan method of equivalence.

math.DG↗