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

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 $ℱ$ 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 $ℱ\left(M\right)$ with respect to $\Gamma$. Our definitions and results generalize the particular cases of the tangent bundle and the tangent bundle of higher order.