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Abstract

We extend the concept of $r$-order connections on fibred manifolds to the one of $(r,s,q)$-order projectable connections on fibred-fibred manifolds, where $r,s,q$ are arbitrary non-negative integers with $s\geq r\leq q$. Similarly to the fibred manifold case, given a bundle functor $F$ of order $r$ on $(m_1,m_2,n_1,n_2)$-dimensional fibred-fibred manifolds $Y\to M$, we construct a general connection $\mathcal{F}({\mit\Gamma},{\mit\Lambda}):FY\to J^1FY$ on $FY\to M$ from a projectable general (i.e. $(1,1,1)$-order) connection ${\mit\Gamma}:Y\to J^{1,1,1}Y$ on $Y\to M$ by means of an $(r,r,r)$-order projectable linear connection ${\mit\Lambda}:TM\to J^{r,r,r}TM$ on $M$. In particular, for $F=J^{1,1,1}$ we construct a general connection $\mathcal{J}^{1,1,1}({\mit\Gamma},\nabla): J^{1,1,1}Y\to J^1J^{1,1,1}Y$ on $J^{1,1,1}Y\to M$ from a projectable general connection ${\mit\Gamma}$ on $Y\to M$ by means of a torsion-free projectable classical linear connection $\nabla$ on $M$. Next, we observe that the curvature of ${\mit\Gamma}$ can be considered as $\mathcal{R}_{\mit\Gamma}:J^{1,1,1}Y\to T^*M\otimes VJ^{1,1,1}Y$. The main result is that if $m_1\geq 2$ and $n_2\geq 1$, then all general connections $D({\mit\Gamma},\nabla):J^{1,1,1}Y\to J^1J^{1,1,1}Y$ on $J^{1,1,1}Y\to M$ canonically depending on ${\mit\Gamma}$ and $\nabla$ form the one-parameter family $\mathcal{J}^{1,1,1}({\mit\Gamma},\nabla)+t\mathcal{R}_{\mit\Gamma}$, $t\in\mathbb{R}$. A similar classification of all general connections $D({\mit\Gamma},\nabla):J^1Y\to J^1J^1Y$ on $J^1Y\to M$ from $({\mit\Gamma}, \nabla)$ is presented.