Jan Slovák

Bernstein-Gelfand-Gelfand sequences

The Bernstein-Gelfand-Gelfand sequences extend the complexes of homogeneous vector bundles to curved Cartan geometries.

Invariant operators on manifolds with almost Hermitian symmetric structures, III. Standard operators

This is the first part of a series of papers. The whole series aims to develop the tools for the study of all almost Hermitian symmetric structures in a unified way. In particular, methods for the construction of invariant operators, their classification and the study of their properties will be worked out. In this paper we present the invariant differentiation with respect to a Cartan connection and we expand the differentials in the terms of the underlying linear connections belonging to the structures in question. Then we discuss the holonomic and non-holonomic jet extensions and we suggest methods for the construction of invariant operators.

Invariant local twistor calculus for quaternionic structures and related geometries

Abstract New universal invariant operators are introduced in a class of geometries which include the quaternionic structures and their generalizations as well as four-dimensional conformal (spin) geometries. It is shown that, in a broad sense, all invariants operators arise from these universal operators and that they may be used to reduce all invariants problems to corresponding algebraic problems involving homomorphisms between modules of certain parabolic subgroups of Lie groups. Explicit application of the operators is illustrated by the construction of all non-standard operators between exterior forms on a large class of the geometries which includes the quaternionic structures.

Morphisms of almost product projective geometries

We discuss almost product projective geometry and the relations to a distinguished class of curves. Our approach is based on an observation that well known general techniques apply, and our goal is to illustrate the power of the general parabolic geometry theory on a quite explicit example. Therefore, some rudiments of the general theory are mentioned on the way, too.

Infinitesimally natural operators are natural

Abstract It is well known that linear geometric operations (like the exterior differential) commute with the Lie derivative. A detailed analysis of both the concepts of geometric operations and of Lie differentiation leads to the proof of a converse implication even in the nonlinear case. So naturality is equivalent to commuting with Lie differentiation. We also generalize this result to the case of gauge-natural operators.

On the equivalence of quaternionic contact structures

Following the Cartan’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 system of differential invariants for two quaternionic contact manifolds to be locally diffeomorphic. This includes an explicit construction of the corresponding Cartan geometry and detailed information on all curvature components.

Matematika drsně a svižně

Zakladni ucebnice matematiky pro vysokoskolske studium. Na MU využivana zejmena jako podpora výuky matematiky na Fakultě informatiky.

Free CR distributions

There are only some exceptional CR dimensions and codimensions such that the geometries enjoy a discrete classification of the pointwise types of the homogeneous models. The cases of CR dimensions n and codimensions n2 are among the very few possibilities of the so-called parabolic geometries. Indeed, the homogeneous model turns out to be PSU(n+1,n)/P with a suitable parabolic subgroup P. We study the geometric properties of such real (2n+n2)-dimensional submanifolds in

DTI Segmentation Using Anisotropy Preserving Quaternion Based Distance Measure.

\mathbb{C}^{n + n^2 }

Building Sustainable R&D Centers in Emerging Technology Regions

for all n > 1. In particular, we show that the fundamental invariant is of torsion type, we provide its explicit computation, and we discuss an analogy to the Fefferman construction of a circle bundle in the hypersurface type CR geometry.