# Detailansicht

Non-standard operators in almost Grassmannian geometry
Ales Navrat
Art der Arbeit
Dissertation
Universität
Universität Wien
Fakultät
Fakultät für Mathematik
Betreuer*in
Andreas Cap
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DOI
10.25365/thesis.24145
URN
urn:nbn:at:at-ubw:1-29108.53173.180665-5
(Print-Exemplar eventuell in Bibliothek verfügbar)

### Abstracts

Abstract
(Deutsch)
An almost Grassmannian geometry of type $(p,q)$ is a parabolic geometry modelled on the Grassmann manifold $Gr_p(\R^{p+q})$ of $p$-dimensional subspaces in $\R^{p+q}$. It is well-known that for $p=q=2$ the structure is equivalent to the four-dimensional conformal geometry. In view of the equivalence, there exists an analogue of the conformally invariant Paneitz operator, transforming functions to densities. This invariant operator is known as the non-standard operator for the almost Grassmannian geometry of type $(2,2)$. In this thesis we deal with almost Grassmannian geometries of type $(2,q)$ with $q>2$. It follows from the general theory of parabolic geometries that the complete obstruction against its local flatness (i.e. isomorphism to the homogeneous model $Gr_2(\R^{2+q})$) are two invariants - a torsion and a curvature. Vanishing of the first one ensures existence of a torsion-free connection. In such a case there exists a family of non-standard invariant operators of order four between exterior forms. We use curved Casimir operators to construct the first of these operators, transforming functions to sections of an irreducible subbundle of four-forms. Next we find a formula for this operator, which is analogous to the formula for Paneitz operator in the sense that it has the same form but contractions with metric are replaced by projections to subbundles of exterior forms. In particular, it shows that the operator factorizes through one-forms and three-forms. The main result of the thesis is theorem \ref{thm}, where we prove that this operator can be extended to an invariant operator on arbitrary almost Grassmannian geometries, including torsion. We also give an explicit formula for this corrected operator and we prove that in the presence of torsion this operator does not factorize as in the torsion-free case.

### Schlagwörter

Schlagwörter
(Deutsch)
parabolic geometry almost Grassmannian geometry invariant differential operator curved Casimir operator non-standard operator
Autor*innen
Ales Navrat
Haupttitel (Englisch)
Non-standard operators in almost Grassmannian geometry
Publikationsjahr
2012
Umfangsangabe
I, 123 S. : graph. Darst.
Sprache
Englisch
Beurteiler*innen
Rod Gover ,