Relationship between $\Gamma (E)\otimes \Gamma (B) $ and $\Gamma(E\otimes B) $

Question

I have a question that I've been thinking about for a while. So If $(E,\pi_E)$ and $(B,\pi_B)$ are some vector bundles over some mainfold $M$. What is the exact relationship between $$\Gamma (E)\otimes \Gamma (B) $$ and $$\Gamma(E\otimes B) $$

Are they the same spaces, or if not, are they at least isomorphic to one another?

Answer 1

The natural map goes this way $\Gamma(E) \otimes \Gamma(F) \to \Gamma(E\otimes F)$.

Injectiveness is not a problem.

Surjectiveness is. Think of compact connected complex manifold, $E$ a bundle with a $0$ space of global sections, while $E\otimes F$ has lots of global sections.

However in the smooth or topological category things are OK because it is true for trivial bundles and it is true for a direct sum if and only if it is true for summands. Moreover, every bundle is a direct summand of a trivial one, like @anomaly: indicated. In fact, bundles are equivalent to finitely generated projective module over the ring of functions on the manifold (Swan's theorem) and things work fine for tensor products.