Each sub-Riemannian geometry with bracket generating distribution enjoys a background structure determined by the distribution itself. At the same time, those geometries with constant sub-Riemannian symbols determine a unique Cartan connection leading to their principal invariants. We provide cohomological description of the structure of these curvature invariants in the cases where the background structure is one of the parabolic geometries. As an illustration, constant curvature models are discussed for certain sub-Riemannian geometries.

We classify all (locally) homogeneous Levi non-degenerate real hypersurfaces in C 3 with symmetry algebra of dimension ≥ 6. Keywords Real hypersurfaces in complex manifolds • Symmetry algebra • Homogeneous • Integrable Legendrian contact structures Mathematics Subject Classification Primary 32V40; Secondary 32V05 • 58J70 • 53A15 Im(w) = z 1z1 ± • • • ± z n−1zn−1 .

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 fivedimensional integrable Legendrian contact structures whose symmetry algebra is transitive on the manifold and has at least one-dimensional isotropy algebra at any point.Regarding c as a function of a, b and treating x, y, u as parameters, we have c a = y + 2b(x + a), c b = (x + a) 2 , and c aa = 2b, c ab = 2(x + a) = ±2 √ c b , c bb = 0.WLOG, the ± ambiguity can be eliminated: the corresponding PDE systems are equivalent via the point transformation (a, b, c) → (−a, b, c). Thus, the dual system to III.6-1 is u 11 = 2y, u 12 = 2 √ q, u 22 = 0.Our classification indicates that III.6-1 is not self-dual (but a priori this is not at all obvious).