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 describe all canonical 2-forms Λ(ω) on the r-th order tangent bundle TrM = Jr0 (R;M) of a symplectic manifold (M, ω). As a corollary we deduce that all canonical symplectic structures Λ(ω) on TrM over a symplectic manifold (M, ω) are of the form Λ(ω) = Σrk=0 αkω(k) for all real numbers αk with αr ≠ 0, where ω(k) is the (k)-lift (in the sense of A. Morimoto) of ω to TrM.

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 outline some of the tools C. Ehresmann introduced in Differential Geometry (fiber bundles, connections, jets, groupoids, pseudogroups). We emphasize two aspects of C. Ehresmann's works: use of Cartan notations for the theory of connections and semi-holonomic jets.

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 $\Lambda$.

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.

We classify all $ℱ{ℳ}_{m,n}$-natural operators $:J²⟿J²V^A$ transforming second order connections Γ: Y → J²Y on a fibred manifold Y → M into second order connections $\left(\Gamma \right):V^{A}Y\to J²V^{A}Y$ on the vertical Weil bundle $V^{A}Y\to M$ corresponding to a Weil algebra A.

For every product preserving bundle functor $T^{\mu }$ on fibered manifolds, we describe the underlying functor of any order $(r,s,q),s\ge r\le q$. We define the bundle $K_{k,l}^{r,s,q}Y$ of $(k,l)$-dimensional contact elements of the order $(r,s,q)$ on a fibered manifold $Y$ and we characterize its elements geometrically. Then we study the bundle of general contact elements of type $\mu$. We also determine all natural transformations of $K_{k,l}^{r,s,q}Y$ into itself and of $T\left(K_{k,l}^{r,s,q}Y\right)$ into itself and we find all natural operators lifting projectable vector fields and horizontal one-forms...