If $E\longrightarrow M$ is a vector bundle then we can associate a new vector bundle $$\Lambda^k E\longrightarrow M$$ whose fiber over $p\in M$ is $\Lambda^k E_p$, that is, the $p$-th exterior power of $E_p$.

Let us denote by $\Omega^k(E)$ the $C^\infty(M)$-module of sections of $\Lambda^k E$.

Take $F\longrightarrow M$ another vector bundle and let $E\oplus F\longrightarrow M$ the whitney sum of $E$ and $F$. Is it true that there is an isomorphism of $C^\infty(M)$-modules $$\Omega^k(E\oplus F)\simeq \bigoplus_{j=0}^k \Omega^{k-j}(E)\otimes_{C^\infty(M)} \Omega^j(F)?$$



Yeah. One way to see it is to use the isomorphisms

$$ \Omega^k(E \oplus F) = \Gamma(\Lambda^k(E \oplus F)) \cong \Lambda^k(\Gamma(E \oplus F)) \cong \Lambda^k(\Gamma(E) \oplus \Gamma(F)) \cong \\ \bigoplus_{j=0}^k \Lambda^j(\Gamma(E)) \otimes_{C^{\infty}(M)} \Lambda^{k - j}(\Gamma(F)) \cong \bigoplus_{j=0}^k \Gamma(\Lambda^j(E)) \otimes_{C^{\infty}(M)} \Gamma(\Lambda^{k-j}(E)) = \\ \bigoplus_{j=0}^k \Omega^j(E) \otimes_{C^{\infty}(M)} \Omega^{k-j}(F). $$

The basic idea is that you can perform any functorial construction on vector bundles (in this case, direct sum and exterior product) and take global sections of the resulting bundle or perform the construction directly on the global sections and the results will be naturally isomorphic. For more details, consult Chapter 7 of "Differentiable Manifolds" by Lawerence Conlon.


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