Embedding a vector bundle into $\mathbb{R}^n$ preserving the structure

Question

I have a rank-$k$ vector bundle $\pi: E \to M$ that is a smooth $m$-dimensional manifold. It can be embedded into $\mathbb{R}^{2m}$ by the virtue of the strong Whitney embedding theorem but I would like to have it additionally preserve its vector bundle structure. Is it possible? I would appreciate some references as I am completely new to this theory and I'm struggling to familiarize myself with it.

Answer 1

Yes, $E$ can be embedded fiberwise inside a trivial bundle $M \times \Bbb R^n$.

For simplicity's sake assume $M$ is compact. Choose a finite cover $\{\mathcal{U}_\alpha\}$ of $M$ such that $E$ is trivial over each $\mathcal{U}_\alpha$ - let's call the trivializations $\varphi_\alpha : p^{-1}(\mathcal{U}_\alpha) \to \mathcal{U}_\alpha \times \Bbb R^k$. $\rho_\alpha$ be a partition of unity on this cover.

Define $f_\alpha : E \to \Bbb R^k$ be the map defined by $f_\alpha(v) = \rho_\alpha(p(v)) \cdot (\pi_2 \circ \varphi_\alpha)$ where $\pi_2$ is the second projection $\mathcal{U}_\alpha \times \Bbb R^k \to \Bbb R^k$. This is a fiberwise injection over $\mathcal U_\alpha$. Thus, look at the product of them all $f : E \to \Bbb R^k \times \cdots \times \Bbb R^k = \Bbb R^n$, that is, $\alpha$-th coordinate of $f(v)$ is defined to be $f_\alpha(v)$ - this is a fiberwise injection on all of $E$.

The desired embedding is then $\iota : E \to M \times \Bbb R^n$ defined by $\iota(v) = (p(v), f(v))$.