Universal Prolongation of Linear Partial Differential Equations on Filtered Manifolds

The aim of this article is to show that systems of linear partial differential equations on filtered manifolds, which are of weighted finite type, can be canonically rewritten as first order systems of a certain type. This leads immediately to obstructions to the existence of solutions. Moreover, we will deduce that the solution space of such equations is always finite dimensional.Comment: 14 pages; v3: minor changes in section 3, typos corrected; final version to appear in Arch. math (Brno) 5, 200

Strongly essential flows on irreducible parabolic geometries

We study the local geometry of irreducible parabolic geometries admitting strongly essential flows; these are flows by local automorphisms with higher-order fixed points. We prove several new rigidity results, and recover some old ones for projective and conformal structures, which show that in many cases the existence of a strongly essential flow implies local flatness of the geometry on an open set having the fixed point in its closure. For almost c-projective and almost quaternionic structures we can moreover show flatness of the geometry on a neighborhood of the fixed point.Comment: 34 pages. Proof of Proposition 3.1 significantly shortened, under slightly less general hypotheses (see Remark 3.1). Typos corrected and references updated. To appear in Transactions of the AM

Cryptic species in tropic sands--interactive 3D anatomy, molecular phylogeny and evolution of meiofaunal Pseudunelidae (Gastropoda, Acochlidia)

Towards realistic estimations of the diversity of marine animals, tiny meiofaunal species usually are underrepresented. Since the biological species concept is hardly applicable on exotic and elusive animals, it is even more important to apply a morphospecies concept on the best level of information possible, using accurate and efficient methodology such as 3D modelling from histological sections. Molecular approaches such as sequence analyses may reveal further, cryptic species. This is the first case study on meiofaunal gastropods to test diversity estimations from traditional taxonomy against results from modern microanatomical methodology and molecular systematics

On automorphism groups of some types of generic distributions

AbstractTo certain types of generic distributions (subbundles in a tangent bundle) one can associate canonical Cartan connections. Many of these constructions fall into the class of parabolic geometries. The aim of this article is to show how strong restrictions on the possibles sizes of automorphism groups of such distributions can be deduced from the existence of canonical Cartan connections. This needs no information on how the Cartan connections are actually constructed and only very basic information on their properties. In particular, we discuss the examples of generic distributions of rank two in dimension five, rank three in dimension six, and rank four in dimension seven

Weighted jet bundles and differential operators for parabolic geometries

Eine filtrierte Mannigfaltigkeit ist eine Mannigfaltigkeit, deren Tangentialbündel eine Filtrierung in Teilvektorbündel besitzt, die mit der Lie Klammer von Vektorfeldern verträglich ist. Studiert man Differentialoperatoren auf filtrierten Mannigfaltigkeiten stellt sich heraus, dass der Begriff der Ordnung eines Differentialoperators an die Filtrierung des Tangentialbündels angepasst werde sollte. Dies führt zu einem Konzept von gewichteten Jetbündeln, das einen geeigneten Rahmen bildet um Differentialgleichungen auf filtrierten Mannigfaltigkeiten zu studieren. In der vorliegenden Arbeit werden Differentialoperatoren zwischen Schnitten von natürlichen Vektorbündeln über bestimmten filtrierten Mannigfaltigkeiten, nämlich über regulären infinitesimalen Flaggenmannigfaltigkeiten, im Rahmen von gewichteten Jetbündeln studiert. Im ersten Teil dieser Arbeit widmen wir uns dem Problem der Prolongation von einer großen Klasse überbestimmter Systeme von Differentialgleichungen auf regulären infinitesimalen Flaggenmannigfaltigkeiten. Im zweiten Teil, werden wir uns mit der Konstruktion von invarianten Differentialoperatoren für parabolische Geometrien mittels gekrümmten Casimir-Operatoren auseinandersetzen und invariante Differentialoperatoren für Lagrange-Kontakt-Strukturen konstuieren.A filtered manifold is a manifold, whose tangent bundle admits a filtration by vectorsubbundles, which is compatible with the Lie bracket of vector fields. Studying differential operators on filtered manifolds, it turns out that the notion of order of a differential operator should be adapted to the filtration of the tangent bundle. This leads to a concept of weighted jet bundles, which forms a convenient framework to investigate differential equations on filtered manifolds. In this thesis, we study differential operators acting between sections of natural vector bundles over certain filtered manifolds, namely over regular infinitesimal flag manifolds, within the framework of weighted jet bundles. In the first part, we deal with the problem of prolongation of a wide class of overdetermined systems of differential equations on regular infinitesimal flag manifolds. In the second part, we are concerned with the construction of invariant operators for parabolic geometries via curved Casimir operators and show how to construct invariant differential operators for Lagrangean contact structures