scholarly journals PROLONGATION ON REGULAR INFINITESIMAL FLAG MANIFOLDS

2012 ◽  
Vol 23 (04) ◽  
pp. 1250007 ◽  
Author(s):  
KATHARINA NEUSSER
Keyword(s):  
Vector Bundle ◽  
Linear Connection ◽  
Solution Space ◽  
Contact Structures ◽  
Flag Manifolds ◽  
Overdetermined System ◽  
Linear Formula ◽  
Parabolic Geometries ◽  
Cr Structures ◽  
Conformal Structures

Many interesting geometric structures can be described as regular infinitesimal flag structures, which occur as the underlying structures of parabolic geometries. Among these structures we have for instance conformal structures, contact structures, certain types of generic distributions and partially integrable almost CR-structures of hypersurface type. The aim of this article is to develop for a large class of (semi-) linear overdetermined systems of partial differential equations on regular infinitesimal flag manifolds M a conceptual method to rewrite these systems as systems of the form [Formula: see text], where [Formula: see text] is a linear connection on some vector bundle V over M and C : V → T* M ⊗ V is a (vector) bundle map. In particular, if the overdetermined system is linear, [Formula: see text] will be a linear connection on V and hence the dimension of its solution space is bounded by the rank of V. We will see that the rank of V can be easily computed using representation theory.

scholarly journals Connections adapted to non-negatively graded structures

2019 ◽  
Vol 16 (02) ◽  
pp. 1950021
Author(s):  
Andrew James Bruce
Keyword(s):  
Vector Bundle ◽  
Geometric Structure ◽  
Vector Bundles ◽  
Natural Generalization ◽  
Lie Algebroids ◽  
Linear Formula ◽  
Natural Sense ◽  
Graded Structures ◽  
Graded Manifolds ◽  
Double Vector Bundle

Graded bundles are a particularly nice class of graded manifolds and represent a natural generalization of vector bundles. By exploiting the formalism of supermanifolds to describe Lie algebroids, we define the notion of a weighted[Formula: see text]-connection on a graded bundle. In a natural sense weighted [Formula: see text]-connections are adapted to the basic geometric structure of a graded bundle in the same way as linear [Formula: see text]-connections are adapted to the structure of a vector bundle. This notion generalizes directly to multi-graded bundles and in particular we present the notion of a bi-weighted[Formula: see text]-connection on a double vector bundle. We prove the existence of such adapted connections and use them to define (quasi-)actions of Lie algebroids on graded bundles.

scholarly journals Isomorphism classes for higher order tangent bundles

2017 ◽  
Vol 17 (2) ◽  
pp. 175-189 ◽  
Author(s):  
Ali Suri
Keyword(s):  
Vector Bundle ◽  
Linear Connection ◽  
Equivalence Classes ◽  
Banach Manifold ◽  
Invariant Vector ◽  
Isomorphism Classes ◽  
Connection Map ◽  
Tangent Bundles ◽  
Order Tangent ◽  
Bundle Structure

AbstractThe tangent bundle TkM of order k of a smooth Banach manifold M consists of all equivalence classes of curves that agree up to their accelerations of order k. In previous work the author proved that TkM, 1 ≤ k ≤∞, admits a vector bundle structure on M if and only if M is endowed with a linear connection, or equivalently if a connection map on TkM is defined. This bundle structure depends heavily on the choice of the connection. In this paper we ask about the extent to which this vector bundle structure remains isomorphic. To this end we define the k-th order differential Tkg : TkM ⟶ TkN for a given differentiable map g between manifolds M and N. As we shall see, Tkg becomes a vector bundle morphism if the base manifolds are endowed with g-related connections. In particular, replacing a connection with a g-related one, where g : M ⟶ M is a diffeomorphism, one obtains invariant vector bundle structures. Finally, using immersions on Hilbert manifolds, convex combinations of connection maps and manifolds of Cr maps we offer three examples for our theory, showing its interaction with known problems such as the Sasaki lift of metrics.

scholarly journals Essential Killing fields of parabolic geometries: projective and conformal structures

2013 ◽  
Vol 11 (12) ◽  
Author(s):  
Andreas Čap ◽  
Karin Melnick
Keyword(s):  
General Theory ◽  
Killing Fields ◽  
Parabolic Geometries ◽  
Conformal Structures ◽  
Conformal Geometries ◽  
Infinitesimal Automorphisms

AbstractWe use the general theory developed in our article [Čap A., Melnick K., Essential Killing fields of parabolic geometries, Indiana Univ. Math. J. (in press)], in the setting of parabolic geometries to reprove known results on special infinitesimal automorphisms of projective and conformal geometries.

Variability of radiosonde-observed precipitable water in the Baltic region*

2005 ◽  
Vol 36 (4-5) ◽  
pp. 423-433 ◽  
Author(s):  
E. Jakobson ◽  
H. Ohvril ◽  
O. Okulov ◽  
N. Laulainen
Keyword(s):  
Water Vapour ◽  
Precipitable Water ◽  
Linear Functions ◽  
Water Vapour Pressure ◽  
Baltic Region ◽  
Geographical Latitude ◽  
Linear Formula ◽  
Annual Means ◽  
The Baltic ◽  
Two Parameters

The total mass of columnar water vapour (precipitable water, W) is an important parameter of atmospheric thermodynamic and radiative models. In this work more than 60 000 radiosonde observations from 17 aerological stations in the Baltic region over 14 years, 1989–2002, were used to examine the variability of precipitable water. A table of monthly and annual means of W for the stations is given. Seasonal means of W are expressed as linear functions of the geographical latitude degree. A linear formula is also derived for parametrisation of precipitable water as a function of two parameters – geographical latitude and surface water vapour pressure.

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