Let $M$ be a smooth $m$-manifold, $p\in M$, $l\in\mathbb{N}$, $\pi:E\to M$ be a smooth vector bundle, $\nabla, \nabla'$ be connections of $E$ and $TM$, respectively. $\Gamma(E)$ denotes the space of smooth sections of $E$. $E_p:=\pi^{-1}(\{p\})\subset E$.
Then we can consider the Whitney sum \begin{eqnarray} \mathbb{E}:=E\oplus(E\otimes T^\ast M)\oplus\cdots\oplus(E\otimes (T^\ast M^{\otimes l})) \end{eqnarray} and the subset \begin{eqnarray} \mathbb{V}_p :=&\{u(p)\oplus\nabla u(p)\oplus \cdots\oplus\nabla^l u(p)\vert u\in\Gamma(E)\}&\subset\mathbb{E}_p, \\ \mathbb{V} :=&\sqcup_{p\in M}\mathbb{V}_p&\subset\mathbb{E}. \end{eqnarray}
Question:
(i) Does $\mathbb{V}$ foam a smooth subbundle of $\mathbb{E}$?
(ii) Suppose $g$ be a smooth Riemannian metric of $M$ and $\nabla'$ be the LeviCivita connection. Does $\mathbb{V}$ depend on $g$ ?
(iii) Are there any simple description of $\mathbb{V}$ in which the arbitrary smooth section $u\in\Gamma(E)$ doesn't appear ?