The infinitesimal symmetry algebra of any Cartan geometry has maximum dimension realized by the flat model, but often this dimension drops significantly when considering non-flat geometries, so a gap phenomenon arises. For general (regular, normal) parabolic geometries of type (G,P), we use Tanaka theory to derive a universal upper bound on the submaximal symmetry dimension. We use Kostant’s version ...

Let Gr(d, n) be the Grassmannian of <i>d</i>-dimensional linear subspaces of an <i>n</i>-dimensional vector space <i>V</i>. A submanifold <i>X</i> ⊂ Gr(<i>d, n</i>) gives rise to a differential system Σ(X) that governs <i>d</i>-dimensional submanifolds of <i>V</i> whose Gaussian image is contained in <i>X</i>. We investigate a special case of this construction where <i>X</i> is a six-fold in Gr(4, ...

In this paper we describe the algebra of differential invariants for GL(n,C)-structures. This leads to classification of almost complex structures of general positions. The invariants are applied to the existence problem of higher-dimensional pseudoholomorphic submanifolds.

We prove involutivity of Einstein and Einstein-Maxwell equations by calculating the Spencer cohomology of these systems. Relation with Cartan method is traced in details. Basic implications through Cartan-Kähler theory are derived.

We review computations of joint invariants on a linear symplectic space, discuss variations for an extension of group and space and relate this to other equivalence problems and approaches, most importantly to differential invariants.

In this paper, we investigate overdetermined systems of scalar PDEs on the plane with one common characteristic, whose general solution depends on one function of one variable. We describe linearization of such systems and their integration via Laplace transformation, relating this to Lie's integration theorem and formal theory of PDEs.

For an almost product structure J on a manifold M of dimension 6 with non-degenerate Nijenhuis tensor N J, we show that the automorphism group G=Aut(M,J) has dimension at most 14. In the case of equality G is the exceptional Lie group G∗2. The next possible symmetry dimension is proved to be equal to 10, and G has Lie algebra sp(4,R). Both maximal and submaximal symmetric structures are globally ...

We discuss rank 2 sub-Riemannian structures on low-dimensional manifolds and prove that some of these structures in dimensions 6, 7 and 8 have a maximal amount of symmetry but no integrals polynomial in momenta of low degrees, except for those coming from the Killing vector fields and the Hamiltonian, thus indicating nonintegrability of the corresponding geodesic flows.

We construct the simplest chaotic system with a two-point attractor.

In 1896 Tresse gave a complete description of relative differential invariants for the pseudogroup action of point transformations on the 2nd order ODEs. The purpose of this paper is to review, in light of modern geometric approach to PDEs, this classification and also discuss the role of absolute invariants and the equivalence problem.