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Affine liftings of torsion-free connections to Weil bundles

100%
Dębecki J.
Colloquium Mathematicum
|
2009
|
tom 114
|
nr 1
1-8
EN
This paper contains a classification of all affine liftings of torsion-free linear connections on n-dimensional manifolds to any linear connections on Weil bundles under the condition that n ≥ 3.
2
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Linear liftings of symmetric tensor fields of type (1,2) to Weil bundles

100%
Dębecki J.
Annales Polonici Mathematici
|
2007
|
tom 92
|
nr 1
13-27
EN
This paper contains a classification of all linear liftings of symmetric tensor fields of type (1,2) on n-dimensional manifolds to any tensor fields of type (1,2) on Weil bundles under the condition that n ≥ 3.
3
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Liftings of forms to Weil bundles and the exterior derivative

100%
Dębecki J.
Annales Polonici Mathematici
|
2009
|
tom 95
|
nr 3
289-300
EN
In a previous paper we have given a complete description of linear liftings of p-forms on n-dimensional manifolds M to q-forms on $T^AM$, where $T^A$ is a Weil functor, for all non-negative integers n, p and q, except the case p = n and q = 0. We now establish formulas connecting such liftings and the exterior derivative of forms. These formulas contain a boundary operator, which enables us to define a homology of the Weil algebra~A. We next study the case p = n and q = 0 under the condition that A is acyclic. Finally, we compute the kernels and the images of the boundary operators for the Weil algebras $𝔻^r_k$ and show that these algebras are acyclic.
4
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On a homology of algebras with unit

100%
Dębecki J.
Annales Polonici Mathematici
|
2014
|
tom 110
|
nr 2
189-208
EN
We present a very general construction of a chain complex for an arbitrary (even non-associative and non-commutative) algebra with unit and with any topology over a field with a suitable topology. We prove that for the algebra of smooth functions on a smooth manifold with the weak topology the homology vector spaces of this chain complex coincide with the classical singular homology groups of the manifold with real coefficients. We also show that for an associative and commutative algebra with unit endowed with the discrete topology this chain complex is dual to the de Rham complex.
5
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Linear liftings of affinors to Weil bundles

100%
Dębecki J.
Colloquium Mathematicum
|
2003
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tom 96
|
nr 2
179-189
EN
We give a classification of all linear natural operators transforming affinors on each n-dimensional manifold M into affinors on $T^{A}M$, where $T^{A}$ is the product preserving bundle functor given by a Weil algebra A, under the condition that n ≥ 2.
6
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Canonical tensor fields of type (p,0) on Weil bundles

100%
Dębecki J.
Annales Polonici Mathematici
|
2006
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tom 88
|
nr 3
271-278
EN
We give a classification of canonical tensor fields of type (p,0) on an arbitrary Weil bundle over n-dimensional manifolds under the condition that n ≥ p. Roughly speaking, the result we obtain says that each such canonical tensor field is a sum of tensor products of canonical vector fields on the Weil bundle.
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