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Free CR distributions

100%
Schmalz G., Slovák J.
Open Mathematics
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2012
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tom 10
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nr 5
1896-1913
EN
There are only some exceptional CR dimensions and codimensions such that the geometries enjoy a discrete classification of the pointwise types of the homogeneous models. The cases of CR dimensions n and codimensions n 2 are among the very few possibilities of the so-called parabolic geometries. Indeed, the homogeneous model turns out to be PSU(n+1,n)/P with a suitable parabolic subgroup P. We study the geometric properties of such real (2n+n 2)-dimensional submanifolds in $\mathbb{C}^{n + n^2 } $ for all n > 1. In particular, we show that the fundamental invariant is of torsion type, we provide its explicit computation, and we discuss an analogy to the Fefferman construction of a circle bundle in the hypersurface type CR geometry.
2
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Cartan connection of transversally Finsler foliation

100%
Miernowski A.
Annales Universitatis Mariae Curie-Skłodowska, sectio A – Mathematica
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2012
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tom 66
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nr 1
EN
The purpose of this paper is to define transversal Cartan connectionof Finsler foliation and to prove its existence and uniqueness.
3
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Subtleties concerning conformal tractor bundles

88%
Graham C., Willse T.
Open Mathematics
|
2012
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tom 10
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nr 5
1721-1732
EN
The realization of tractor bundles as associated bundles in conformal geometry is studied. It is shown that different natural choices of principal bundle with normal Cartan connection corresponding to a given conformal manifold can give rise to topologically distinct associated tractor bundles for the same inducing representation. Consequences for homogeneous models and conformal holonomy are described. A careful presentation is made of background material concerning standard tractor bundles and equivalence between parabolic geometries and underlying structures.
4
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Explicit expression of Cartan’s connection for Levi-nondegenerate 3-manifolds in complex surfaces, and identification of the Heisenberg sphere

75%
Merker J., Sabzevari M.
Open Mathematics
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2012
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tom 10
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nr 5
1801-1835
EN
We study effectively the Cartan geometry of Levi-nondegenerate C 6-smooth hypersurfaces M 3 in ℂ2. Notably, we present the so-called curvature function of a related Tanaka-type normal connection explicitly in terms of a graphing function for M, which is the initial, single available datum. Vanishing of this curvature function then characterizes explicitly the local biholomorphic equivalence of such M 3 ⊂ ℂ2 to the Heisenberg sphere ℍ3, such M’s being necessarily real analytic.
5
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Distinguished geodesics and jacobi fields on first order jet spaces

51%
Balan V., Voicu N.
Open Mathematics
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2004
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tom 2
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nr 4
516-526
EN
In the framework of jet spaces endowed with a non-linear connection, the special curves of these spaces (h-paths, v-paths, stationary curves and geodesics) which extend the corresponding notions from Riemannian geometry are characterized. The main geometric objects and the paths are described and, in the case when the vertical metric is independent of fiber coordinates, the first two variations of energy and the extended Jacobi field equations are derived.
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