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

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 $$$$C^\infty(U,V_1\otimes\cdots\otimes V_n)\to\Gamma(U,E_1\otimes\cdots\otimes E_n)$$$$ 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 $$$$\Phi_m\colon V_1\otimes\cdots\otimes V_n\to(E_1)_m\otimes\cdots\otimes(E_n)_m$$$$ to be the unique isomorphism s.t. $$\Phi_m(v_1\otimes\cdots\otimes v_n)=[s_1(m),v_1]\otimes\cdots\otimes[s_n(m),v_n]$$.

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

$$^1$$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).

• What exactly is the question? Your $\Phi$ seems to me like a good guess... Jun 7, 2021 at 15:21
• @nicrot000 Since it was only a guess, I wished to have a confirmation. Jun 7, 2021 at 19:26

## 1 Answer

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 $$v\mapsto[s_i(m),v]$$.

• Please let me know if I understand your answer correctly: Consider a vector bundle $\pi\colon E\to M$ with general fiber $V$ and a bundle atlas $A$. Each bundle chart $\phi\colon\pi^{-1}(U)\to U\times V$ in $A$ is bijective and therefore defines a bijection $\Gamma(U,E)\to\Gamma(U,M\times V)$. Since $\Gamma(U,M\times V)$ is isomorphic to the set of functions from $U$ to $V$, a bundle chart allows us to identify local sections with vector valued functions. My question is about the special case where $E$ is the tensor product bundle of associated vector bundles. Jun 8, 2021 at 14:31
• It doesn't matter if $V$ is just a generic space, or if $V$ is assumed to be a space of the shape $V=V_1\otimes...\otimes V_n$ with some other spaces $V_1,\dots,V_n$. I think I can't point out where exactly your confusion lies in. Jun 8, 2021 at 17:42
• For the first part, yes this is correct, up to the detail that the bijectivity of $\phi$ alone is not enough, you really need that $\phi$ is an isomorphism of the bundles $\pi^{-1}(U)$ and $U\times V$. Jun 8, 2021 at 17:46