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

J. Kurek; W. M. Mikulski

Displaying similar documents to “Lifting adapted connections from foliated manifolds to higher order adapted frame bundles”

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.

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.

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.

Neifeld’s Connection Inducedon the Grassmann Manifold

Olga Belova (2016)

Acta Universitatis Palackianae Olomucensis. Facultas Rerum Naturalium. Mathematica

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The work concerns to investigations in the field of differential geometry. It is realized by a method of continuations and scopes of G. F. Laptev which generalizes a moving frame method and Cartan’s exterior forms method and depends on calculation of exterior differential forms. The Grassmann manifold (space of all $m$ -planes) is considered in the $n$ -dimensional projective space $P_{n}$ . Principal fiber bundle of tangent linear frames is arised above this manifold. Typical fiber of the principal...

Non-existence of some canonical constructions on connections

Włodzimierz M. Mikulski (2003)

Commentationes Mathematicae Universitatis Carolinae

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For a vector bundle functor $H : ℳ f \to 𝒱 ℬ$ with the point property we prove that $H$ is product preserving if and only if for any $m$ and $n$ there is an $ℱ ℳ_{m, n}$ -natural operator $D$ transforming connections $Γ$ on $(m, n)$ -dimensional fibered manifolds $p : Y \to M$ into connections $D (Γ)$ on $H p : H Y \to H M$ . For a bundle functor $E : ℱ ℳ_{m, n} \to ℱ ℳ$ with some weak conditions we prove non-existence of $ℱ ℳ_{m, n}$ -natural operators $D$ transforming connections $Γ$ on $(m, n)$ -dimensional fibered manifolds $Y \to M$ into connections $D (Γ)$ on $E Y \to M$ .

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.

A construction of a connection on $G Y \to Y$ from a connection on $Y \to M$ by means of classical linear connections on $M$ and $Y$

Włodzimierz M. Mikulski (2005)

Commentationes Mathematicae Universitatis Carolinae

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Let $G$ be a bundle functor of order $(r, s, q)$ , $s \geq r \leq q$ , on the category $ℱ ℳ_{m, n}$ of $(m, n)$ -dimensional fibered manifolds and local fibered diffeomorphisms. Given a general connection $Γ$ on an $ℱ ℳ_{m, n}$ -object $Y \to M$ we construct a general connection $𝒢 (Γ, λ, Λ)$ on $G Y \to Y$ be means of an auxiliary $q$ -th order linear connection $λ$ on $M$ and an $s$ -th order linear connection $Λ$ on $Y$ . Then we construct a general connection $𝒢 (Γ, \nabla_{1}, \nabla_{2})$ on $G Y \to Y$ by means of auxiliary classical linear connections $\nabla_{1}$ on $M$ and $\nabla_{2}$ on $Y$ . In the case $G = J^{1}$ we determine all general connections...

The natural operators lifting vector fields to generalized higher order tangent bundles

Włodzimierz M. Mikulski (2000)

Archivum Mathematicum

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For natural numbers $r$ and $n$ and a real number $a$ we construct a natural vector bundle $T^{(r), a}$ over $n$ -manifolds such that $T^{(r), 0}$ is the (classical) vector tangent bundle $T^{(r)}$ of order $r$ . For integers $r \geq 1$ and $n \geq 3$ and a real number $a < 0$ we classify all natural operators $T_{| M_{n}} ⇝ T T^{(r), a}$ lifting vector fields from $n$ -manifolds to $T^{(r), a}$ .

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 $M$ .