Tensor product vector bundle - bijection from vector valued functions to sections defined by local sections

Question

Let $G_i\to P_i\to M$ be principal fiber bundles with representations $\rho_i\colon G_i\to\mathrm{GL}(V_i)$ and associated vector bundles $E_i\to M$. Given local sections $s_i\colon U\to P_i$, I expect that a bijection \begin{equation} C^\infty(U,V_1\otimes\cdots\otimes V_n)\to\Gamma(U,E_1\otimes\cdots\otimes E_n) \end{equation} can be constructed.

Here's my guess:

For all $m\in U\subset M$, \begin{align} V_i&\to (E_i)_m\\ v&\mapsto[s_i(m),v] \end{align} is a bijection$^1$ and we define \begin{equation} \Phi_m\colon V_1\otimes\cdots\otimes V_n\to(E_1)_m\otimes\cdots\otimes(E_n)_m \end{equation} to be the unique isomorphism s.t. $\Phi_m(v_1\otimes\cdots\otimes v_n)=[s_1(m),v_1]\otimes\cdots\otimes[s_n(m),v_n]$.

The isomorphism \begin{equation} \Phi\colon C^\infty(U,V_1\otimes\cdots\otimes V_n)\to\Gamma(U,E_1\otimes\cdots\otimes E_n) \end{equation} is then defined by \begin{equation} (\Phi(f))(m)=(\Phi_m\circ f)(m). \end{equation}

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$^1$This follows from the definition of the equivalence classes and the fact that $G\ni g\to pg\in P$ is a bijection for all $p\in M$ (according to the definition of principal fiber bundles).

Answer 1

So, since you asked in the comments above, here is your confirmation.

Maybe I can add, that since you have sections $s_i$ on $U$ in each of your principal bundles, they are all trivializable over that same $U$, consequently each of the associated vector bundles is trivializable over this $U$.

On the other hand, $C^\infty(U,V_1\otimes\cdots\otimes V_n)$ is a section space over $U$ in the trivial bundle $U\times(V_1\otimes\cdots\otimes V_n)$ and applying the inverse of the respective trivialization to each factor should simply give you the isomorphism $$ C^\infty(U,V_1\otimes\cdots\otimes V_n)\to\Gamma(U,E_1\otimes\cdots\otimes E_n). $$ The (inverse) trivilization in each factor is then, as you remarked in the footnote, given by $v\mapsto[s_i(m),v]$.