2,811 research outputs found
Preferred Parameterisations on Homogeneous Curves
We show how to specify preferred parameterisations on a homogeneous curve in
an arbitrary homogeneous space. We apply these results to limit the natural
parameters on distinguished curves in parabolic geometries.Comment: 10 page
Conformally Fedosov manifolds
We introduce the notion of a conformally Fedosov structure and construct an
associated Cartan connection. When an appropriate curvature vanishes, this
allows us to construct a family of natural differential complexes akin to the
BGG complexes from parabolic geometry.Comment: 28 pages. This is a substantial update to include BGG machinery and
the construction of differential complexe
Calculus on symplectic manifolds
On a symplectic manifold, there is a natural elliptic complex replacing the
de Rham complex. It can be coupled to a vector bundle with connection and, when
the curvature of this connection is constrained to be a multiple of the
symplectic form, we find a new complex. In particular, on complex projective
space with its Fubini-Study form and connection, we can build a series of
differential complexes akin to the Bernstein-Gelfand-Gelfand complexes from
parabolic differential geometry.Comment: 17 page
Bernstein-Gelfand-Gelfand sequences
This paper is devoted to the study of geometric structures modeled on
homogeneous spaces G/P, where G is a real or complex semisimple Lie group and
is a parabolic subgroup. We use methods from differential geometry
and very elementary finite-dimensional representation theory to construct
sequences of invariant differential operators for such geometries, both in the
smooth and the holomorphic category. For G simple, these sequences specialize
on the homogeneous model G/P to the celebrated (generalized)
Bernstein-Gelfand-Gelfand resolutions in the holomorphic category, while in the
smooth category we get smooth analogs of these resolutions. In the case of
geometries locally isomorphic to the homogeneous model, we still get
resolutions, whose cohomology is explicitly related to a twisted de Rham
cohomology. In the general (curved) case we get distinguished curved analogs of
all the invariant differential operators occurring in Bernstein-Gelfand-Gelfand
resolutions (and their smooth analogs).
On the way to these results, a significant part of the general theory of
geometrical structures of the type described above is presented here for the
first time.Comment: 45 page
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