Natural transformations of affinors into linear forms
Natural transformations of Lagrangians
Natural transformations of affinors into functions and affinors
Affine liftings of torsion-free connections to Weil bundles
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.
Linear liftings of affinors to Weil bundles
We give a classification of all linear natural operators transforming affinors on each n-dimensional manifold M into affinors on , where is the product preserving bundle functor given by a Weil algebra A, under the condition that n ≥ 2.
Linear liftings of skew symmetric tensor fields of type to Weil bundles
The paper contains a classification of linear liftings of skew symmetric tensor fields of type on -dimensional manifolds to tensor fields of type on Weil bundles under the condition that It complements author’s paper “Linear liftings of symmetric tensor fields of type to Weil bundles” (Ann. Polon. Math. , 2007, pp. 13–27), where similar liftings of symmetric tensor fields were studied. We apply this result to generalize that of author’s paper “Affine liftings of torsion-free connections...
On a homology of algebras with unit
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...
Canonical tensor fields of type (p,0) on Weil bundles
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.
Liftings of forms to Weil bundles and the exterior derivative
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 , where 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....
Linear liftings of symmetric tensor fields of type (1,2) to Weil bundles
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.
Linear liftings of skew-symmetric tensor fields to Weil bundles
We define equivariant tensors for every non-negative integer and every Weil algebra and establish a one-to-one correspondence between the equivariant tensors and linear natural operators lifting skew-symmetric tensor fields of type on an -dimensional manifold to tensor fields of type on if . Moreover, we determine explicitly the equivariant tensors for the Weil algebras , where and are non-negative integers.
Some liftings of Poisson structures to Weil bundles
We establish a formula for the Schouten-Nijenhuis bracket of linear liftings of skew-symmetric tensor fields to any Weil bundle. As a result we obtain a construction of some liftings of Poisson structures to Weil bundles.
The natural operators transforming affinors to tensor fields of type
Linear natural operators lifting -vectors to tensors of type on Weil bundles
We give a classification of all linear natural operators transforming -vectors (i.e., skew-symmetric tensor fields of type ) on -dimensional manifolds to tensor fields of type on , where is a Weil bundle, under the condition that , and . The main result of the paper states that, roughly speaking, each linear natural operator lifting -vectors to tensor fields of type on is a sum of operators obtained by permuting the indices of the tensor products of linear natural operators lifting...
Invariant vector fields of Hamiltonians
A complete classification of natural transformations of Hamiltonians into vector fields on symplectic manifolds is given herein.
Natural transformations of symplectic structures into Poisson's and Jacobi's brackets
A complete classification of natural transformations of symplectic structures into Poisson's brackets as well as into Jacobi's brackets is given.
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