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I'v asked this question on StackExchange but unfortunately nobody answered. I thought that maybe it would be more apropriate to post it here:

so suppose that we have a morphism $\theta: \Gamma(B,E_1) \to \Gamma(B,E_2)$ where $E_i$ are two vector bundles over $B$ and let $f:A \to B$ be a continuos map. Then we can define a pullback bundles $f^*E_1$ and $f^*E_2$.

Question 1: first of all I would like to know whether one can somehow describe $\Gamma(A,f^*E_1)$ in terms of $\Gamma(B,E_1)$

And then I would also like to know:

Question 2: How a given morphism $\theta$ like above induces morphism $f^*\theta: \Gamma(A,f^*E_1) \to \Gamma(A,f^*E_2)$.

I tried the following argumentation: once we have a morphism $\theta$ like above it produces a morphism of bundles $\varphi:E_1 \to E_2$. This induces a morphism $(x,v) \mapsto (x,\varphi(v))$ between pullback bundles and this gives rise to the morphism on the level of sections. Is there a direct approach, which allows us working only on the level of sections? If so, is it equivalent to described above?

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  • $\begingroup$ Use projection formula. $\endgroup$
    – Sasha
    Apr 21, 2015 at 21:22
  • $\begingroup$ 1) I don't think you can get anything more explicit than the formula you get using the universal property. Same for 2), except if you are in a special situation. Your approach works if and only if the vector bundle is completely determined by its global sections. This isn't always the case, consider for example the Möbius bundle over the unit circle. $\endgroup$ Apr 23, 2015 at 7:43

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