Related papers: Complete complex parabolic geometries
We investigate geometric properties of homogeneous parabolic geometries with generalized symmetries. We show that they can be reduced to a simpler geometric structures and interpret them explicitly. For specific types of parabolic…
First we introduce a generalization of symmetric spaces to parabolic geometries. We provide construction of such parabolic geometries starting with classical symmetric spaces and we show that all regular parabolic geometries with smooth…
We generalize the concept of locally symmetric spaces to parabolic contact structures. We show that symmetric normal parabolic contact structures are torsion--free and some types of them have to be locally flat. We prove that each symmetry…
The current paper is devoted to the study of integral curves of constant type in parabolic homogeneous spaces. We construct a canonical moving frame bundle for such curves and give the criterium when it turns out to be a Cartan connection.…
We construct a series of examples of non--flat non--homogeneous parabolic geometries that carry a symmetry of the parabolic geometry at each point.
In this article, we summarize the results on symmetric conformal geometries. We review the results following from the general theory of symmetric parabolic geometries and prove several new results for symmetric conformal geometries. In…
The general theory of parabolic geometries is applied to the study of the normal Cartan connections for all hyperbolic and elliptic 6-dimensional CR-manifolds of codimension two. The geometric meaning of the individual components of the…
The first examples of complete projective connections are uncovered: normal projective connections on surfaces whose geodesics are all closed and embedded are complete, as are normal projective connections induced from complete affine…
We classify the holomorphic parabolic geometries on compact complex manifolds of general type. We accomplish this by bounding the numerical dimension of any smooth projective variety in terms of geometric invariants of the flag variety…
We investigate (local) automorphisms of parabolic geometries that generalize geodesic symmetries. We show that many types of parabolic geometries admit at most one generalized geodesic symmetry at a point with non-zero harmonic curvature.…
We consider automorphisms of homogeneous parabolic geometries with a fixed point. Parabolic geometries carry the distinguished distributions and we study those automorphisms which enjoy natural actions on the distributions at the fixed…
Hyperbolic geometry is developed in a purely algebraic fashion from first principles, without a prior development of differential geometry. The natural connection with the geometry of Lorentz, Einstein and Minkowski comes from a projective…
This article describes an entirely algebraic construction for developing conformal geometries, which provide models for, among others, the Euclidean, spherical and hyperbolic geometries. On one hand, their relationship is usually shown…
In any dimension at least five we construct examples of closed smooth manifolds with the following properties: 1) they have neither real projective nor flat conformal structures; 2) their fundamental group is a non-elementary Gromov…
We prove that the only complex parabolic geometries on Calabi-Yau manifolds are the homogeneous geometries on complex tori. We also classify the complex parabolic geometries on homogeneous compact K\"ahler manifolds.
A special linear Lie group over the real number field and the quarternion field admits a projectivley flat affine connection. We show that parabolic subgroups are autoparallel submanifolds and give a criterion the induced connection is…
All parabolic geometries, i.e. Cartan geometries with homogeneous model a real generalized flag manifold, admit highly interesting classes of distinguished curves. The geodesics of a projective class of connections on a manifold, conformal…
The following results are proved: Theorem 1. A totally real semiparallel submanifold of constant curvature with parallel f-structure in the normal bundle of a K\"ahler manifold N is flat or a totally geodesic submanifold of N. Theorem 2. A…
We generalize the concept of affine locally symmetric spaces for parabolic geometries. We discuss mainly $|1|$--graded geometries and we show some restrictions on their curvature coming from the existence of symmetries. We use the theory of…
In the physics literature, Bilal--Fock--Kogan \cite{BFK} introduced the idea of parabolic reduced flat connections on a surface to give a geometric origin to $W$-algebras. In this paper, we combine these ideas with higher complex…