Tomas Salac

Tomas Salac
Charles University in Prague | CUNI · Department of Mathematics and Mathematical Education

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19
Publications
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67
Citations

Publications

Publications (19)
Article
Full-text available
This paper gives a uniform, self-contained, and direct approach to a variety of obstruction-theoretic problems on manifolds of dimension 7 and 6. We give necessary and sufficient cohomological criteria for the existence of various G-structures on vector bundles over such manifolds especially using low dimensional representations of U(2)\documentcla...
Preprint
Full-text available
This paper gives a uniform, self-contained and direct approach to a variety of obstruction-theoretic problems on manifolds of dimension 7 and 6. We give necessary and sufficient cohomological criteria for the existence of various G-structures on vector bundles over such manifolds especially using low dimensional representations of the group U(2).
Preprint
This is the first part in a series of three articles in which are studied the domains of monogenicity for the $n$-Cauchy-Fueter operator. Using the twistor theory, we will in this article show that for a given open subset $U$ of $\mathbb{Q}^n$, there is an open subset $\mathcal{H}(U)$, called the monogenic hull of $U$, of $M_{2n\times 2}^{\mathbb{C...
Article
This is the second part in a series of two papers. The k-Dirac complex is a complex of differential operators which are naturally associated to a particular |2|-graded parabolic geometry. In this paper we will consider the k-Dirac complex over the homogeneous space of the parabolic geometry and as a first result, we will prove that the k-Dirac comp...
Article
This is the second part in a series of two papers. The $k$-Dirac complex is a complex of differential operators which are natural to a particular $|2|$-graded parabolic geometry. In this paper we will consider the $k$-Dirac complex over a homogeneous space of the parabolic geometry and as a first result, we will prove that the $k$-Dirac complex is...
Article
This is a first paper in a series of two papers. In this paper we construct complexes (which we call the $k$-Dirac complexes) of invariant differential operators which live on homogeneous spaces of $|2|$-graded parabolic geometries of some particular type. More explicitly we will show that these complexes arise as the direct images of relative BGG...
Article
Full-text available
The Grassmannian G2+(Rn+2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ G_2^+(\mathbb {R}^{n+2})$$\end{document} of oriented 2-planes in Rn+2\documentclass[12pt]{min...
Article
Full-text available
This is the last part of a series of articles on a family of geometric structures (PACS-structures) which all have an underlying almost conformally symplectic structure. While the first part of the series was devoted to the general study of these structures, the second part focused on the case that the underlying structure is conformally symplectic...
Article
The k-Dirac operator is a first order differential operator which is natural to a particular class of parabolic geometries which include the Lie contact structures. A natural task is to understand the set of local null solutions of the operator at a given point. We will show that this set has a very nice and simple structure, namely we will show th...
Article
Full-text available
Parabolic almost conformally symplectic structures were introduced in the first part of this series of articles as a class of geometric structures which have an underlying almost conformally symplectic structure. If this underlying structure is conformally symplectic, then one obtains a PCS-structure. In the current article, we relate PCS-structure...
Article
Full-text available
We introduce a class of first order G-structures, each of which has an underlying almost conformally symplectic structure. There is one such structure for each real simple Lie algebra which is not of type $C_n$ and admits a contact grading. We show that a structure of each of these types on a smooth manifold $M$ determines a canonical compatible li...
Article
The k-Dirac operator is a differential operator which is natural to geometric structure of a parabolic type. We will give a set of initial conditions for this operator. In the proof of the claim we will need to adapt some parts from the theory of exterior differential systems to the setting of weighted differential operators.
Article
Given a contact manifold M#M# together with a transversal infinitesimal automorphism ξ , we show that any local leaf space M for the foliation determined by ξ naturally carries a conformally symplectic (cs-) structure. Then we show that the Rumin complex on M#M# descends to a complex of differential operators on M, whose cohomology can be computed....
Article
Full-text available
Given a contact manifold $M_\#$ together with a transversal infinitesimal automorphism $\xi$, we show that any local leaf space $M$ for the foliation determined by $\xi$ naturally carries locally conformally symplectic (lcs--) structure. Then we show that the Rumin complex on $M_\#$ descends to a complex of differential operators on $M$, whose coho...
Article
We apply the Cartan-Kahler theorem for the k-Dirac operator studied in Clifford analysis and to the parabolic version of this operator. We show that for k = 2 the tableaux of the first prolongations of these two operators are involutive. This gives us a new characterization of the set of initial conditions for the 2-Dirac operator.
Article
The Hartog’s type phenomena in several complex variables are best understood in terms of the Dolbeault sequence. A lot of attention was paid in the last decades to its analogue in the function theory of several Clifford variables, i.e. the Dirac operator in several variables. A so-called BGG resolution of this operator is then an analogue to the D...
Article
Penrose transform tells us that there is an isomorphism of the kernel of an invariant di?erential operator studied in the paper [TS] and sheaf cohomology of some vector bundle on twistor space. The point of this paper is to write down this isomorphism explicitly. Explicit form of the isomorphism will be crucial for further investigation on the prop...
Article
The principal group of a Klein geometry has canonical left action on the homogeneous space of the geometry and this action induces action on the spaces of sections of vector bundles over the homogeneous space. This paper is about construction of differential operators invariant with respect to the induced action of the principal group of a particul...
Article
A 3-form ω∈Λ³R7⁎ is called multisymplectic if it satisfies some natural non-degeneracy requirement. It is well known that there are 8 orbits (or types) of multisymplectic 3-forms on R⁷ under the canonical action of GL(7,R) and that two types are open. This leads to 8 types of global multisymplectic 3-forms on 7-dimensional manifolds without boundar...

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