SCHWEITZER 1985 Problem 11

Let $\xi (E, \pi, B) (\pi\colon E\rightarrow B)$ be a real vector bundle of finite rank, and let $$\tau_E=V\xi \oplus H\xi\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, (*)$$ be the tangent bundle of $E$ , where $V\xi=\mathrm{Ker} d\pi$ is the vertical subbundle of $\tau_E$ . Let us denote the projection operators corresponding to the splitting $(*)$ by $v$ and $h$ . Construct a linear connection $\nabla$ on $V\xi$ such that $$\nabla_X\lor Y - \nabla_Y \lor X=v[X,Y] - v[hX,hY]$$ ( $X$ and $Y$ are vector fields on $E$ , $[., .]$ is the Lie bracket, and all data are of class $\mathcal C^\infty$ . [J. Szilasi]