In what follows all manifolds, Lie groups and mappings are meant to be $C^\infty$.
Let $\pi:M\longrightarrow B$ be a left $G$-principal bundle. A connection on this bundle is a map $H$ which assigns to every $p\in M$ a vector subspace $H_p\subseteq T_pM$ such that:
$(i)$ $T_pM=H_p\oplus V_p$ (where $V_p$ are the vertical tangent vectors to $p$);
$(ii)$ $(dR_g)_p(H_p)=H_{g\cdot p}$ (I'm not sure if this is right, the notation of the paper I'm following is outdated so I'm writing here what I think it makes sense);
$(iii)$ $H$ is differentiable: Given any smooth vector field $X$ on $M$ if $X^h$ denotes its horizontal part then $X^h$ is smooth.
Can I realize $H$ as a section of a vector bundle (with additional properties)?
Thanks