Andreas Čap
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View article: BGG Sequences -- A Riemannian perspective
BGG Sequences -- A Riemannian perspective Open
BGG resolutions and generalized BGG resolutions from representation theory of semisimple Lie algebras have been generalized to sequences of invariant differential operators on manifolds endowed with a geometric structure belonging to the f…
View article: On Relative Tractor Bundles
On Relative Tractor Bundles Open
This article contributes to the relative BGG-machinery for parabolic geometries. Starting from a relative tractor bundle, this machinery constructs a sequence of differential operators that are naturally associated to the geometry in quest…
View article: Partial AHS-Structures, their Cartan description and partial BGG sequences
Partial AHS-Structures, their Cartan description and partial BGG sequences Open
G-structures and Cartan geometries are two major approaches to the description of geometric structures (in the sense of differential geometry) on manifolds of some fixed dimension $n$. We show that both descriptions naturally extend to the…
View article: Flat extensions of principal connections and the Chern-Simons $3$-form
Flat extensions of principal connections and the Chern-Simons $3$-form Open
We introduce the notion of a flat extension of a connection $θ$ on a principal bundle. Roughly speaking, $θ$ admits a flat extension if it arises as the pull-back of a component of a Maurer-Cartan form. For trivial bundles over closed orie…
View article: A Boundary-Local Mass Cocycle and the Mass of Asymptotically Hyperbolic Manifolds
A Boundary-Local Mass Cocycle and the Mass of Asymptotically Hyperbolic Manifolds Open
We construct a cocycle that, for a given n -manifold, maps a pair of asymptotically locally hyperbolic (ALH) metrics to a tractor-valued $$(n-1)$$ -form field on the conformal infinity. This requires the metrics to be asymptotically…
View article: On Relative Tractor Bundles
On Relative Tractor Bundles Open
This article contributes to the relative BGG-machinery for parabolic geometries. Starting from a relative tractor bundle, this machinery constructs a sequence of differential operators that are naturally associated to the geometry in quest…
View article: Poisson transforms, the BGG complex, and discrete series representations of SU(n+1,1)
Poisson transforms, the BGG complex, and discrete series representations of SU(n+1,1) Open
The aim of this article is to construct a specific Poisson transform mapping differential forms on the sphere $S^{2n+1}$ endowed with its natural CR structure to forms on complex hyperbolic space. The transforms we construct have values th…
View article: BGG Sequences with Weak Regularity and Applications
BGG Sequences with Weak Regularity and Applications Open
We investigate some Bernstein–Gelfand–Gelfand complexes consisting of Sobolev spaces on bounded Lipschitz domains in $${\mathbb {R}}^{n}$$ . In particular, we compute the cohomology of the conformal deformation complex and the conform…
View article: Bounded Poincaré operators for twisted and BGG complexes
Bounded Poincaré operators for twisted and BGG complexes Open
We construct bounded Poincaré operators for twisted complexes and BGG complexes with a wide class of function classes (e.g., Sobolev spaces) on bounded Lipschitz domains. These operators are derived from the de Rham versions using BGG diag…
View article: Induced almost para-Kähler Einstein metrics on cotangent bundles
Induced almost para-Kähler Einstein metrics on cotangent bundles Open
In earlier work we have shown that for certain geometric structures on a smooth manifold $M$ of dimension $n$, one obtains an almost para-Kähler--Einstein metric on a manifold $A$ of dimension $2n$ associated to the structure on $M$. The g…
View article: Bundles of Weyl structures and invariant calculus for parabolic geometries
Bundles of Weyl structures and invariant calculus for parabolic geometries Open
For more than hundred years, various concepts were developed to understand the fields of geometric objects and invariant differential operators between them for conformal Riemannian and projective geometries. More recently, several general…
View article: Geometric theory of Weyl structures
Geometric theory of Weyl structures Open
Given a parabolic geometry on a smooth manifold [Formula: see text], we study a natural affine bundle [Formula: see text], whose smooth sections can be identified with Weyl structures for the geometry. We show that the initial parabolic ge…
View article: BGG sequences with weak regularity and applications
BGG sequences with weak regularity and applications Open
We investigate some Bernstein-Gelfand-Gelfand (BGG) complexes on bounded Lipschitz domains in $\mathbb{R}^n$ consisting of Sobolev spaces. In particular, we compute the cohomology of the conformal deformation complex and the conformal Hess…
View article: A relative mass cocycle and the mass of asymptotically hyperbolic manifolds
A relative mass cocycle and the mass of asymptotically hyperbolic manifolds Open
We construct a cocycle that, for a given $n$-manifold, maps pairs of asymptotically locally hyperbolic (ALH) metrics to a tractor-valued $(n-1)$-form field on the conformal infinity. This requires the metrics to be asymptotically related t…
View article: A Poisson transform adapted to the Rumin complex
A Poisson transform adapted to the Rumin complex Open
Let G be a semisimple Lie group with finite center, [Formula: see text] a maximal compact subgroup, and [Formula: see text] a parabolic subgroup. Following ideas of P. Y. Gaillard, one may use G-invariant differential forms on [Formula: se…
View article: C<sup>1</sup> deformations of almost-Grassmannian structures with strongly essential symmetry
C<sup>1</sup> deformations of almost-Grassmannian structures with strongly essential symmetry Open
We construct a family of $(2,n)$-almost Grassmannian structures of regularity $C^1$, each admitting a one-parameter group of strongly essential automorphisms, and each not flat on any neighborhood of the higher-order fixed point. This show…
View article: Parabolic Conformally Symplectic Structures. III: Invariant Differential Operators and Complexes
Parabolic Conformally Symplectic Structures. III: Invariant Differential Operators and Complexes Open
This is the last part of a series of articles on a family of geometric structures (PACS-structures) which all have an underlying almost conformally symplectic structure. While the first part of the series was devoted to the general study o…
View article: c-projective compactification; (quasi-)Kähler metrics and CR boundaries
c-projective compactification; (quasi-)Kähler metrics and CR boundaries Open
For complete complex connections on almost complex manifolds we introduce a\nnatural definition of compactification. This is based on almost c--projective\ngeometry, which is the almost complex analogue of projective differential\ngeometry…
View article: On C-class equations
On C-class equations Open
The concept of a C-class of differential equations goes back to E. Cartan with the upshot that generic equations in a C-class can be solved without integration. While Cartan's definition was in terms of differential invariants being first …
View article: On canonical Cartan connections associated to filtered G-structures
On canonical Cartan connections associated to filtered G-structures Open
A filtered manifold is a smooth manifold $M$ together with a filtration of the tangent bundle by smooth subbundles which is compatible with the Lie bracket of vector fields in a certain sense. The Lie bracket of vector fields then induces …
View article: Parabolic conformally symplectic structures III; Invariant differential operators and complexes
Parabolic conformally symplectic structures III; Invariant differential operators and complexes Open
This is the last part of a series of articles on a family of geometric structures (PACS-structures) which all have an underlying almost conformally symplectic structure. While the first part of the series was devoted to the general study o…