The vertical prolongation of the projectable connections

Abstract

We prove that any first order $\mathcal{F}_2\mathcal{M}_{m_1,m_2,n_1,n_2}$-natural operator transforming projectable general connections on an $(m_1,m_2, n_1, n_2)$-dimensional fibred-fibred manifold $p = (p, p) : (p_Y : Y \to Y ) \to (p_M : M \to M)$ into general connections on the vertical prolongation $V Y \to M$ of $p: Y \to M$ is the restriction of the (rather well-known) vertical prolongation operator $\mathcal{V}$ lifting general connections $\overline{\Gamma}$ on a fibred manifold $Y \to M$ into $\mathcal{V}\overline{\Gamma}$ (the vertical prolongation of $\overline{\Gamma}$) on $V Y \to M$.