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1
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Connections in regular Poisson manifolds over ℝ-Lie foliations

100%
Kubarski J.
Banach Center Publications
|
2000
|
tom 51
|
nr 1
141-149
EN
The subject of this paper is the notion of the connection in a regular Poisson manifold M, defined as a splitting of the Atiyah sequence of its Lie algebroid. In the case when the characteristic foliation F is an ℝ-Lie foliation, the fibre integral operator along the adjoint bundle is used to define the Euler class of the Poisson manifold M. When M is oriented 3-dimensional, the notion of the index of a local flat connection with singularities along a closed transversal is defined. If, additionally, F has compact leaves (then F is a fibration over $S^{1}$), an analogue of the Euler-Poincaré-Hopf index theorem for flat connections with singularities along closed transversals is obtained.
2
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Algebroid nature of the characteristic classes of flat bundles

100%
Kubarski J.
Banach Center Publications
|
1998
|
tom 45
|
nr 1
199-224
EN
The following two homotopic notions are important in many domains of differential geometry: - homotopic homomorphisms between principal bundles (and between other objects), - homotopic subbundles. They play a role, for example, in many fundamental problems of characteristic classes. It turns out that both these notions can be - in a natural way - expressed in the language of Lie algebroids. Moreover, the characteristic homomorphisms of principal bundles (the Chern-Weil homomorphism [K4], or the subject of this paper, the characteristic homomorphism for flat bundles) are invariants of Lie algebroids of these bundles. This enables one to construct the characteristic homomorphism of a flat regular Lie algebroid, measuring the incompatibility of the flat structure with a given subalgebroid. For two given Lie subalgebroids, these homomorphisms are equivalent if the Lie subalgebroids are homotopic. Some new examples of applications of this characteristic homomorphism to a transitive case (for TC-foliations) and to a non-transitive case (for a principal bundle equipped with a partial flat connection) are pointed out (Ex. 3.1). An example of a transitive Lie algebroid of a TC-foliation which leads to the nontrivial characteristic homomorphism is obtained.
3
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Linear direct connections

64%
Kubarski J., Teleman N.
Banach Center Publications
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2007
|
tom 76
|
nr 1
425-436
EN
In this paper we study the geometry of direct connections in smooth vector bundles (see N. Teleman [Tn.3]); we show that the infinitesimal part, $∇^{τ}$, of a direct connection τ is a linear connection. We determine the curvature tensor of the associated linear connection $∇^{τ}.$ As an application of these results, we present a direct proof of N. Teleman's Theorem 6.2 [Tn.3], which shows that it is possible to represent the Chern character of smooth vector bundles as the periodic cyclic homology class of a specific periodic cyclic cycle $Φ_{*}^{τ},$ manufactured from a direct connection τ, rather than from a smooth linear connection as the Chern-Weil construction does. In addition, we show that the image of the cyclic cycle $Φ_{*}^{τ}$ into the de Rham cohomology (through the A. Connes' isomorphism) coincides with the cycle provided by the Chern-Weil construction applied to the underlying linear connection $∇^{τ}.$ For more details about these constructions, the reader is referred to [M], N. Teleman [Tn.1], [Tn.2], [Tn.3], C. Teleman [Tc], A. Connes [C.1], [C.2] and A. Connes and H. Moscovici [C.M].
4
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Nondegenerate cohomology pairing for transitive Lie algebroids, characterization

64%
Kubarski J., Mishchenko A.
Open Mathematics
|
2004
|
tom 2
|
nr 5
663-707
EN
The Evens-Lu-Weinstein representation (Q A, D) for a Lie algebroid A on a manifold M is studied in the transitive case. To consider at the same time non-oriented manifolds as well, this representation is slightly modified to (Q Aor, Dor) by tensoring by orientation flat line bundle, Q Aor=QA⊗or (M) and D or=D⊗∂Aor. It is shown that the induced cohomology pairing is nondegenerate and that the representation (Q Aor, Dor) is the unique (up to isomorphy) line representation for which the top group of compactly supported cohomology is nontrivial. In the case of trivial Lie algebroid A=TM the theorem reduce to the following: the orientation flat bundle (or (M), ∂Aor) is the unique (up to isomorphy) flat line bundle (ξ, ∇) for which the twisted de Rham complex of compactly supported differential forms on M with values in ξ possesses the nontrivial cohomology group in the top dimension. Finally it is obtained the characterization of transitive Lie algebroids for which the Lie algebroid cohomology with trivial coefficients (or with coefficients in the orientation flat line bundle) gives Poincaré duality. In proofs of these theorems for Lie algebroids it is used the Hochschild-Serre spectral sequence and it is shown the general fact concerning pairings between graded filtered differential ℝ-vector spaces: assuming that the second terms live in the finite rectangular, nondegeneration of the pairing for the second terms (which can be infinite dimensional) implies the same for cohomology spaces.
5
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Complete differentials of higher order in linear field modules

44%
Kubarski J.
Annales Polonici Mathematici
|
1984
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tom 44
|
nr 2
131-153
6
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Exponential mapping for Lie groupoids. Applications

38%
Kubarski J.
Colloquium Mathematicum
|
1987
|
tom 54
|
nr 1
39-48
7
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Exponential mapping for Lie groupoids

32%
Kubarski J.
Colloquium Mathematicum
|
1982
|
tom 47
|
nr 2
267-282
8
Content available remote

Preface

26%
Kubarski J., Rybicki T., Wolak R.
Open Mathematics
|
2004
|
tom 2
|
nr 5
i-iii
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