The Theory of Matrices. By F R. Gantmacher. Two volumes, pp. 374 and 276. 1959. (Translated from the Russian by K. A. Hirsch; Chelsea Publishing Company, New York)

Introduction to lie algebras and representation theory

Sub-Riemannian geometry is the geometry of a world with nonholonomic constraints. In such a world, one can move, send and receive information only in certain admissible directions but eventually can reach every position from any other. In the last two decades sub-Riemannian geometry has emerged as an independent research domain impacting on several areas of pure and applied mathematics, with applications to many areas such as quantum control, Hamiltonian dynamics, robotics and Lie theory. This comprehensive introduction proceeds from classical topics to cutting-edge theory and applications, assuming only standard knowledge of calculus, linear algebra and differential equations. The book may serve as a basis for an introductory course in Riemannian geometry or an advanced course in sub-Riemannian geometry, covering elements of Hamiltonian dynamics, integrable systems and Lie theory. It will also be a valuable reference source for researchers in various disciplines.

https://hal.archives-ouvertes.fr/hal-02019181/document

A Comprehensive Introduction to Sub-Riemannian Geometry

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Transformation Groups In Differential Geometry

/pdf/geometric-structures-on-filtered-manifolds-1vf9vuo3ek.pdf

Geometric structures on filtered manifolds

On the integrability of symplectic Monge–Ampère 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.

/pdf/generalized-wilczynski-invariants-for-non-linear-ordinary-3eve423qyu.pdf

Generalized Wilczynski Invariants for Non-Linear Ordinary Differential Equations

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).

https://mosaic.math.tamu.edu/~zelenko/comptlongarx.pdf

On local geometry of non-holonomic rank 2 distributions

Homogeneous surfaces in the three-dimensional affine geometry

The current paper is devoted to the study of integral curves of constant type in parabolic homogeneous spaces. We con- struct a canonical moving frame bundle for such curves and give the criterium when it turns out to be a Cartan connection. Generaliza- tions to parametrized curves, to higher-dimensional submanifolds and to general parabolic geometries are discussed. This paper is devoted to the local geometry of curves in parabolic homogeneous spaces. We provide the uniform approach to this prob- lem based on algebraic properties of so-called distinguished curves and their symmetry algebras. In particular, we develop the methods that allow to deal with any curves without any additional non-degeneracy assumptions. Unlike the classical moving frame methods and techniques (9, 16, 15, 18, 19, 20), we avoid branchings as much as possible treating all curves at once. In fact, the only branching used in our paper is done at a very first step based on the type of the tangent line to a curve, or, in other words on its 1-st jet. For each curve type there is a class of most symmetric curves, also known as distinguished curves in parabolic geometry (7). There are sim- ple algorithms to compute the symmetry algebra of these curves (10). We try to approximate any curve by the distinguished curves of the same type and construct the smallest possible moving frame bundle for all curves of a given type. It has the same dimension as the symmetry algebra of the distinguished curve, and its normalization conditions as well as the number of fundamental invariants (or curvatures) is defined by the simple algebraic properties of this symmetry algebra. We are mainly interested in curves that are integral curves of the vec- tor distribution defined naturally on any parabolic homogeneous space.

Boris Doubrov

Papers

On the integrability of symplectic Monge–Ampère equations

Generalized Wilczynski Invariants for Non-Linear Ordinary Differential Equations

On local geometry of non-holonomic rank 2 distributions

Homogeneous surfaces in the three-dimensional affine geometry

Geometry of curves in parabolic homogeneous spaces