Report on K-theory for convenient algebras
Two constructions with parabolic geometries
Natural operators between vector valued differential forms
[For the entire collection see Zbl 0742.00067.]This paper is devoted to a method permitting to determine explicitly all multilinear natural operators between vector-valued differential forms and between sections of several other natural vector bundles.
Report on K-theory for convenient algebras. II.
Automorphism groups of parabolic geometries
Some special geometry in dimension six
Motivated by the study of CR-submanifolds of codimension in , the authors consider here a -dimensional oriented manifold equipped with a -dimensional distribution. Under some non-degeneracy condition, two different geometric situations can occur. In the elliptic case, one constructs a canonical almost complex structure on ; the hyperbolic case leads to a canonical almost product structure. In both cases the only local invariants are given by the obstructions to integrability for these structures....
On local flatness of manifolds with AHS-structures
Summary: The AHS-structures on manifolds are the simplest cases of the so called parabolic geometries which are modeled on homogeneous spaces corresponding to a parabolic subgroup in a semisimple Lie group. It covers the cases where the negative parts of the graded Lie algebras in question are abelian. In the series the authors developed a consistent frame bundle approach to the subject. Here we give explicit descriptions of the obstructions against the flatness of such structures based on the latter...
Characteristic classes for -bundles
The authors generalize a construction of Connes by defining for an -bundle over smooth manifold and a reduced cyclic cohomology class a sequence of de Rham cohomology classes . Here is a convenient algebra, defined by the authors, and is a locally trivial bundle with standard fibre a right finitely generated projective -module and bounded -modules homomorphisms as transition functions.
Curved Casimir operators and the BGG machinery.
Infinitesimal automorphisms and deformations of parabolic geometries
We show that infinitesimal automorphisms and infinitesimal deformations of parabolic geometries can be nicely described in terms of the twisted de Rham sequence associated to a certain linear connection on the adjoint tractor bundle. For regular normal geometries, this description can be related to the underlying geometric structure using the machinery of BGG sequences. In the locally flat case, this leads to a deformation complex, which generalizes the well known complex for locally conformally...
Translation of natural operators on manifolds with AHS-structures
We introduce an explicit procedure to generate natural operators on manifolds with almost Hermitian symmetric structures and work out several examples of this procedure in the case of almost Grassmannian structures.
All linear and bilinear natural concomitants of vector valued differential forms
On left invariant CR structures on
There is a well known one–parameter family of left invariant CR structures on . We show how purely algebraic methods can be used to explicitly compute the canonical Cartan connections associated to these structures and their curvatures. We also obtain explicit descriptions of tractor bundles and tractor connections.
Essential Killing fields of parabolic geometries: projective and conformal structures
We 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.
Editors’ preface for the topical issue “Finite dimensional integrable systems, dynamics, and Lie theoretic methods in Geometry and Mathematical Physics”
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