The Frölicher-Nijenhuis bracket on some functional spaces

Ivan Kolář; Marco Modungo

Displaying similar documents to “The Frölicher-Nijenhuis bracket on some functional spaces”

New versions of curvature and torsion formulas for the complete lifting of a linear connection to Weil bundles

A. Ntyam, J. Wouafo Kamga (2003)

Annales Polonici Mathematici

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New versions of Slovák’s formulas expressing the covariant derivative and curvature of the linear connection $_{A} Γ$ are presented.

Connections of higher order and product preserving functors

Jacek Gancarzewicz, Noureddine Rahmani, Modesto R. Salgado (2002)

Czechoslovak Mathematical Journal

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In this paper we consider a product preserving functor $ℱ$ of order $r$ and a connection $Γ$ of order $r$ on a manifold $M$ . We introduce horizontal lifts of tensor fields and linear connections from $M$ to $ℱ (M)$ with respect to $Γ$ . Our definitions and results generalize the particular cases of the tangent bundle and the tangent bundle of higher order.

On "special" fibred coordinates for general and classical connections

Włodzimierz M. Mikulski (2010)

Annales Polonici Mathematici

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Using a general connection Γ on a fibred manifold p:Y → M and a torsion free classical linear connection ∇ on M, we distinguish some “special” fibred coordinate systems on Y, and then we construct a general connection $\tilde{ℱ} (Γ, \nabla)$ on Fp:FY → FM for any vector bundle functor F: ℳ f → of finite order.

Natural affinors on the (r,s,q)-cotangent bundle of a fibered manifold

J. Kurek, W. M. Mikulski (2004)

Annales Polonici Mathematici

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For natural numbers r,s,q,m,n with s ≥ r ≤ q we describe all natural affinors on the (r,s,q)-cotangent bundle $T^{r, s, q *} Y$ over an (m,n)-dimensional fibered manifold Y.

Lifting adapted connections from foliated manifolds to higher order adapted frame bundles

J. Kurek, W. M. Mikulski (2008)

Annales Polonici Mathematici

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Let (M,ℱ) be a foliated manifold. We describe all natural operators lifting ℱ-adapted (i.e. projectable in adapted coordinates) classical linear connections ∇ on (M,ℱ) into classical linear connections (∇) on the rth order adapted frame bundle $P^{r} (M, ℱ)$ .

On quasijet bundles

Tomáš, Jiří

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In this paper a Weil approach to quasijets is discussed. For given manifolds $M$ and $N$ , a quasijet with source $x \in M$ and target $y \in N$ is a mapping $T_{x}^{r} M \to T_{y}^{r} N$ which is a vector homomorphism for each one of the $r$ vector bundle structures of the iterated tangent bundle $T^{r}$ [, Casopis Pest. Mat. 111, No. 4, 345-352 (1986; Zbl 0611.58004)]. Let us denote by $Q J^{r} (M, N)$ the bundle of quasijets from $M$ to $N$ ; the space ${\tilde{J}}^{r} (M, N)$ of non-holonomic $r$ -jets from $M$ to $N$ is embeded into $Q J^{r} (M, N)$ . On the other hand, the bundle $Q T_{m}^{r} N$ of $(m, r)$ -quasivelocities...

On the fiber product preserving gauge bundle functors on vector bundles

Włodzimierz M. Mikulski (2003)

Annales Polonici Mathematici

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We present a complete description of all fiber product preserving gauge bundle functors F on the category $_{m}$ of vector bundles with m-dimensional bases and vector bundle maps with local diffeomorphisms as base maps. Some corollaries of this result are presented.

Non-existence of natural operators transforming connections on $Y \to M$ into connections on $F Y \to Y$

Włodzimierz M. Mikulski (2005)

Archivum Mathematicum

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Under some weak assumptions on a bundle functor $F : F M_{m, n} \to F M$ we prove that there is no $F M_{m, n}$ -natural operator transforming connections on $Y \to M$ into connections on $F Y \to Y$ .

Natural maps depending on reductions of frame bundles

Ivan Kolář (2011)

Annales Polonici Mathematici

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We clarify how the natural transformations of fiber product preserving bundle functors on $ℱ ℳ_{m}$ can be constructed by using reductions of the rth order frame bundle of the base, $ℱ ℳ_{m}$ being the category of fibered manifolds with m-dimensional bases and fiber preserving maps with local diffeomorphisms as base maps. The iteration of two general r-jet functors is discussed in detail.

Bundle functors with the point property which admit prolongation of connections

W. M. Mikulski (2010)

Annales Polonici Mathematici

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Let F:ℳ f →ℱℳ be a bundle functor with the point property F(pt) = pt, where pt is a one-point manifold. We prove that F is product preserving if and only if for any m and n there is an $ℱ ℳ_{m, n}$ -canonical construction D of general connections D(Γ) on Fp:FY → FM from general connections Γ on fibred manifolds p:Y → M.

Trivial noncommutative principal torus bundles

Stefan Wagner (2011)

Banach Center Publications

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A (smooth) dynamical system with transformation group ⁿ is a triple (A,ⁿ,α), consisting of a unital locally convex algebra A, the n-torus ⁿ and a group homomorphism α: ⁿ → Aut(A), which induces a (smooth) continuous action of ⁿ on A. In this paper we present a new, geometrically oriented approach to the noncommutative geometry of trivial principal ⁿ-bundles based on such dynamical systems, i.e., we call a dynamical system (A,ⁿ,α) a trivial noncommutative principal ⁿ-bundle if each isotypic...

The natural operators lifting connections to higher order cotangent bundles

Włodzimierz M. Mikulski (2014)

Czechoslovak Mathematical Journal

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We prove that the problem of finding all $ℳ f_{m}$ -natural operators $C : Q ⇝ Q T^{r *}$ lifting classical linear connections $\nabla$ on $m$ -manifolds $M$ into classical linear connections $C_{M} (\nabla)$ on the $r$ -th order cotangent bundle $T^{r *} M = J^{r} {(M, ℝ)}_{0}$ of $M$ can be reduced to the well known one of describing all $ℳ f_{m}$ -natural operators $D : Q ⇝ ⨂^{p} T \otimes ⨂^{q} T^{*}$ sending classical linear connections $\nabla$ on $m$ -manifolds $M$ into tensor fields $D_{M} (\nabla)$ of type $(p, q)$ on $M$ .

Product preserving gauge bundle functors on vector bundles

Włodzimierz M. Mikulski (2001)

Colloquium Mathematicae

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A complete description is given of all product preserving gauge bundle functors F on vector bundles in terms of pairs (A,V) consisting of a Weil algebra A and an A-module V with $d i m_{ℝ} (V) < \infty$ . Some applications of this result are presented.