Wyniki wyszukiwania - Biblioteka Nauki

1

Lie tensor product manifolds

100%

Sato H. , Yamaguchi K.

Demonstratio Mathematica

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2012

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tom Vol. 45, nr 4

909-927

EN

We study the geometric structures of parabolic geometries. A parabolic geometry is defined by a parabolic subgroup of a simple Lie group corresponding to a subset of the positive simple roots. We say that a parabolic geometry is fundamental if it is defined by a subset corresponding to a single simple root. In this paper we will be mainly concerned with such fundamental parabolic geometries. Fundamental geometries for the Lie algebra of (…) type are Grassmann structures. For (…) types, we investigate the geometric feature of the fundamental geometries modeled after the quotients of the real simple groups of split type by the parabolic subgroups. We name such geometries Lie tensor product structures. Especially, we call Lie tensor metric structure for (…) or (…) type and Lie tensor symplectic structure for (…) type. For each manifold with a Lie tensor product structure, we give a unique normal Cartan connection by the method due to Tanaka. Invariants of the structure are the curvatures of the connection.

2

Geometry of t-projective structures

100%

Miernowski A. , Mozgawa W.

Demonstratio Mathematica

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2002

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tom Vol. 35, nr 2

391-403

EN

The main purpose of this paper is to define and study t-projective structure on even dimensional manifold M as a certain reduction of the second order frame bundle over M. This t-structure reveals some similarities to the projective structures of Kobayashi-Nagano [KN1] but it is a completely different one. The structure group is the isotropy group of the tangent bundle of the projective space. With the t-structure we associate in a natural way the so called normal Cartan connection and we investigate its properties. We show that t-structures are closely related the almost tangent structures on M. Finally, we consider the natural cross sections and we derive the coefficients of the normal connection of a t-projective structure.

3

Cartan connection of transversally Finsler foliation

80%

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.

4

Free CR distributions

80%

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.

5

Subtleties concerning conformal tractor bundles

70%

Graham C. , Willse T.

Open Mathematics

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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.

6

Explicit expression of Cartan’s connection for Levi-nondegenerate 3-manifolds in complex surfaces, and identification of the Heisenberg sphere

60%

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.

7

Distinguished geodesics and jacobi fields on first order jet spaces

41%

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.