### A complement to the paper “On the Kolář connection” [Arch. Math. (Brno) 49 (2013), 223–240]

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Let $G$ be a bundle functor of order $(r,s,q)$, $s\ge r\le q$, on the category $\mathcal{F}{\mathcal{M}}_{m,n}$ of $(m,n)$-dimensional fibered manifolds and local fibered diffeomorphisms. Given a general connection $\Gamma $ on an $\mathcal{F}{\mathcal{M}}_{m,n}$-object $Y\to M$ we construct a general connection $\mathcal{G}(\Gamma ,\lambda ,\Lambda )$ on $GY\to Y$ be means of an auxiliary $q$-th order linear connection $\lambda $ on $M$ and an $s$-th order linear connection $\Lambda $ on $Y$. Then we construct a general connection $\mathcal{G}(\Gamma ,{\nabla}_{1},{\nabla}_{2})$ on $GY\to Y$ 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 $\mathcal{D}(\Gamma ,\nabla )$ on ${J}^{1}Y\to Y$ from...

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.

Weil algebra morphisms induce natural transformations between Weil bundles. In some well known cases, a natural transformation is endowed with a canonical structure of affine bundle. We show that this structure arises only when the Weil algebra morphism is surjective and its kernel has null square. Moreover, in some cases, this structure of affine bundle passes to jet spaces. We give a characterization of this fact in algebraic terms. This algebraic condition also determines an affine structure...

We prove that any bundle functor F:ℱol → ℱℳ on the category ℱ olof all foliated manifolds without singularities and all leaf respecting maps is of locally finite order.

Let $(M,\mathcal{F})$ be a foliated $m+n$-dimensional manifold $M$ with $n$-dimensional foliation $\mathcal{F}$. Let $V$ be a finite dimensional vector space over $\mathbf{R}$. We describe all canonical ($\mathcal{F}{\mathit{\text{ol}}}_{m,n}$-invariant) $V$-valued $1$-forms $\Theta :T{P}^{r}(M,\mathcal{F})\to V$ on the $r$-th order adapted frame bundle ${P}^{r}(M,\mathcal{F})$ of $(M,\mathcal{F})$.

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.

Let Y be a fibered square of dimension (m1, m2, n1, n2). Let V be a finite dimensional vector space over. We describe all 21,m2,n1,n2 - canonical V -valued 1-form Θ TPrA (Y) → V on the r-th order adapted frame bundle PrA(Y).

We consider a vector bundle $E\to M$ and the principal bundle $PE$ of frames of $E$. Let $K$ be a principal connection on $PE$ and let $\Lambda $ be a linear connection on $M$. We classify all principal connections on ${W}^{2}PE={P}^{2}M{\times}_{M}{J}^{2}PE$ naturally given by $K$ and $\Lambda $.

This paper is a continuation of [2], dealing with a general, not-necessarily torsion-free, connection. It characterizes all possible systems of generators for vector-field valued operators that depend naturally on a set of vector fields and a linear connection, describes the size of the space of such operators and proves the existence of an ‘ideal’ basis consisting of operators with given leading terms which satisfy the (generalized) Bianchi–Ricci identities without corrections.

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

Let be M a smooth manifold, A a local algebra and ${M}^{A}$ a manifold of infinitely near points on M of kind A. We build the canonical foliation on ${M}^{A}$ and we show that the canonical foliation on the tangent bundle TM is the foliation defined by its canonical field.

We introduce the concept of modified vertical Weil functors on the category $\mathcal{F}{\mathcal{M}}_{m}$ of fibred manifolds with $m$-dimensional bases and their fibred maps with embeddings as base maps. Then we describe all fiber product preserving bundle functors on $\mathcal{F}{\mathcal{M}}_{m}$ in terms of modified vertical Weil functors. The construction of modified vertical Weil functors is an (almost direct) generalization of the usual vertical Weil functor. Namely, in the construction of the usual vertical Weil functors, we replace the usual Weil...

We describe the fiber product preserving bundle functors on the category of all morphisms of fibered manifolds in terms of infinite sequences of Weil algebras and actions of the skeleton of the category of $r$-jets by algebra homomorphisms. We deduce an explicit formula for the iteration of two such functors. We characterize the functors with values in vector bundles.

For every Lie groupoid Φ with m-dimensional base M and every fiber product preserving bundle functor F on the category of fibered manifolds with m-dimensional bases and fiber preserving maps with local diffeomorphisms as base maps, we construct a Lie groupoid ℱ Φ over M. Every action of Φ on a fibered manifold Y → M is extended to an action of ℱ Φ on FY → M.

We derive both local and global generalized Bianchi identities for classical Lagrangian field theories on gauge-natural bundles. We show that globally defined generalized Bianchi identities can be found without the a priori introduction of a connection. The proof is based on a global decomposition of the variational Lie derivative of the generalized Euler-Lagrange morphism and the representation of the corresponding generalized Jacobi morphism on gauge-natural bundles. In particular, we show that...

We generalize reduction theorems for classical connections to operators with values in k-th order natural bundles. Using the 2nd order valued reduction theorems we classify all (0,2)-tensor fields on the cotangent bundle of a manifold with a linear (non-symmetric) connection.