Related papers: Paraconformal structures, ordinary differential eq…

Motivated by recent works in Levi degenerate CR geometry, this article endeavours to study the wider and more flexible para-CR structures for which the constraint of invariancy under complex conjugation is relaxed. We consider…

The author has elsewhere given a complete classification of those compact oriented Einstein 4-manifolds on which the self-dual Weyl curvature is everywhere positive in the direction of some self-dual harmonic 2-form. In this article,…

Following the Cartans's original method of equivalence supported by methods of parabolic geometry, we provide a complete solution for the equivalence problem of quaternionic contact structures, that is, the problem of finding a complete…

A conic of the Veronese surface in PG(5,3) is a quadrangle. If one such quadrangle is replaced with its diagonal triangle, then one obtains a point model $K$ for Witt's 5-$(12,6,1)$ design, the blocks being the hyperplane sections…

We investigate the integrability of natural almost complex structures on the twistor space of an almost para-quaternionic manifold as well as the integrability of natural almost paracomplex structures on the reflector space of an almost…

We introduce and discuss (local) symmetries of geometric structures. These symmetries generalize the classical (locally) symmetric spaces to various other geometries. Our main tools are homogeneous Cartan geometries and their explicit…

We develop methods for constructing and computing conformal invariants of submanifolds, with a particular emphasis on conformal submanifold scalars and conformally invariant integrals of natural submanifold scalars. These methods include a…

We consider the nonstandard inclusion of SO(3) in SO(5) associated with a 5-dimensional irreducible representation. The tensor $\Upsilon$ representing this reduction is found to be given by a ternary symmetric form with special properties.…

$2$-nondegenerate real hypersurfaces in complex manifolds play an important role in CR-geometry and the theory of Hermitian Symmetric Domains. In this paper, we construct a complete convergent normal form for everywhere $2$-nondegenerate…

For several classes of second order dispersionless PDEs, we show that the symbols of their formal linearizations define conformal structures which must be Einstein-Weyl in 3D (or self-dual in 4D) if and only if the PDE is integrable by the…

We reconsider non-degenerate second order superintegrable systems in dimension two as geometric structures on conformal surfaces. This extends a formalism developed by the authors, initially introduced for (pseudo-)Riemannian manifolds of…

The fundamental structure of the 4-dimensional spacetime is assumed to be the lorentzian CR-structure (LCR-structure), which contains two correlated 3-dimensional CR-structures. It is defined by explicit Frobenius integrable relations…

We study $GL(2)$-structures on differential manifolds. The structures play a fundamental role in the geometric theory of ordinary differential equations. We prove that any $GL(2)$-structure on an even dimensional manifold give rise to a…

Twistor correspondences for R-invariant indefinite self-dual conformal structures on R^4 are established explicitly. These correspondences are written down by using a natural integral transform from functions on a two dimensional cylinder…

Riemannian geometry in four dimensions naturally leads to an SL(3) connection that annihilates a basis for self-dual two-forms. Einstein's equations may be written in terms of an SO(3) connection, with SO(3) chosen as an appropriate…

We summarize some our recent results on encoding exact solutions of field equations in Einstein and modified gravity theories into solitonic hierarchies derived for nonholonomic curve flows with associated bi-Hamilton structure. We argue…

A finite subgroup of the conformal group SL(2,C) can be related to invariant polynomials on a hypersurface in C^3. The latter then carries a simple singularity, which resolves by a finite iteration of basic cycles of deprojections. The…

For a four-dimensional (nonisoclinicly geodesic) three-web W (3, 2, 2), a transversal distribution $\Delta$ is defined by the torsion tensor of the web. In general, this distribution is not integrable. The authors find necessary and…

By referring to theorems of Donaldson and Hitchin, we exhibit a rigorous AdS/CFT-type correspondence between classical 2+1 dimensional vacuum general relativity theory on S x R and SO(3) Hitchin theory (regarded as a classical conformal…

This paper is concerned with the construction of special metrics on non-compact 4-manifolds which arise as resolutions of complex orbifold singularities. Our study is close in spirit to the construction of the hyperkaehler gravitational…