879 research outputs found
CR-Tractors and the Fefferman Space
We develop the natural tractor calculi associated to conformal and
CR structures as a fundamental tool for the study of Fefferman's construction
of a canonical conformal class on the total space of a circle bundle over a
non--degenerate CR manifold of hypersurface type. In particular we construct
and treat the basic objects that relate the natural bundles and natural
operators on the two spaces. This is illustrated with several applications: We
prove that a number of conformally invariant overdetermined systems admit
non--trivial solutions on any Fefferman space. We show that the space of
conformal Killing fields on a Fefferman space naturally decomposes into a
direct sum of subspaces, which admit an interpretaion as solutions of certain
CR invariant PDE's. Finally we explicitly analyze the relation between tractor
calculus on a CR manifold and the complexified conformal tractor calculus on
its Fefferman space, thus obtaining a powerful computational tool for working
with the Fefferman construction.Comment: AMSLaTeX, 46 pages, v3: added link
http://www.iumj.indiana.edu/IUMJ/fulltext.php?year=2008&volume=57&artid=3359
to published version, which has different numbering of statement
Projective BGG equations, algebraic sets, and compactifications of Einstein geometries
For curved projective manifolds we introduce a notion of a normal tractor
frame field, based around any point. This leads to canonical systems of
(redundant) coordinates that generalise the usual homogeneous coordinates on
projective space. These give preferred local maps to the model projective space
that encode geometric contact with the model to a level that is optimal, in a
suitable sense. In terms of the trivialisations arising from the special
frames, normal solutions of classes of natural linear PDE (so-called first BGG
equations) are shown to be necessarily polynomial in the generalised
homogeneous coordinates; the polynomial system is the pull back of a polynomial
system that solves the corresponding problem on the model. Thus questions
concerning the zero locus of solutions, as well as related finer geometric and
smooth data, are reduced to a study of the corresponding polynomial systems and
algebraic sets. We show that a normal solution determines a canonical manifold
stratification that reflects an orbit decomposition of the model. Applications
include the construction of structures that are analogues of Poincare-Einstein
manifolds.Comment: 22 page
Holonomy reductions of Cartan geometries and curved orbit decompositions
We develop a holonomy reduction procedure for general Cartan geometries. We
show that, given a reduction of holonomy, the underlying manifold naturally
decomposes into a disjoint union of initial submanifolds. Each such submanifold
corresponds to an orbit of the holonomy group on the modelling homogeneous
space and carries a canonical induced Cartan geometry. The result can therefore
be understood as a `curved orbit decomposition'. The theory is then applied to
the study of several invariant overdetermined differential equations in
projective, conformal and CR-geometry. This makes use of an equivalent
description of solutions to these equations as parallel sections of a tractor
bundle. In projective geometry we study a third order differential equation
that governs the existence of a compatible Einstein metric. In CR-geometry we
discuss an invariant equation that governs the existence of a compatible
K\"{a}hler-Einstein metric.Comment: v2: major revision; 30 pages v3: final version to appear in Duke
Math.
Conformally Invariant Operators via Curved Casimirs: Examples
We discuss a general scheme for a construction of linear conformally
invariant differential operators from curved Casimir operators; we then
explicitly carry this out for several examples. Apart from demonstrating the
efficacy of the approach via curved Casimirs, this shows that this method
applies both in regular and in singular infinitesimal character, and also that
it can be used to construct standard as well as non--standard operators. The
examples treated include conformally invariant operators with leading term, in
one case, a square of the Laplacian, and in another case, a cube of the
Laplacian.Comment: AMSLaTeX, 16 pages, v2: minor changes, final version to appear in
Pure Appl. Math.
Solution of ordinary differential equations by means of Lie series
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Lie series for celestial mechanics, accelerators, satellite stabilization and optimization
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