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Given a vector bundle $(E,M,\pi,F)$, my notes states that the set $\Gamma(M,E)$ of sections of the bundle is a module over $C^\infty(M)$.

My question is about the operations that make $\Gamma(M,E)$ a module. I dint find much information about this.

My guess is that for $s$ in $\Gamma(M,E)$ then $\pi(s(p))=p$ for all p in $M$, and this is equivalent to $s(p)$ in $\pi^{-1}(p)$, and we know $\pi^{-1}(p)$ is a vector space, so we use the operation on this space, say $+$, to define an operation in $\Gamma(M,E)$.

For example: if $s_1$, $s_2$ in $\Gamma(M,E)$, then $(s_1+s_2)(p)=s_1(p)+s_2(p)$, and in this case $s_i(p)$ in $\pi^{-1}(p)$, so $(s_1+s_2)(p)$ in $\pi^{-1}(p)$.

Is this correct? Any reference about this definition?. Thanks.

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Yes, you essentially have it right. The module structure is a result of point-wise addition of smooth sections $\pi: E \to M$. Multiplication is over the space $C^\infty(M)$, where we have that if $s_1, s_2 \in \Gamma(M,E)$ and $f \in C^\infty(M)$ we have that we obtain a new section $s_1 + fs_2 \in \Gamma(M,E)$ which is defined pointwise by

$$ (s_1 + fs_2)(p) \;\; =\;\; s_1(p) + f(p) s_2(p) $$

for all $p\in M$. The pointwise definition allows us to see that for each $p \in M$ that $(s_1 + fs_2)(p)$ is in the fiber of $\pi:E \to M$ over $p$ (denoted $\pi^{-1}\{p\}$) which is simply just a vector space. Since $\pi$ maps each $v \in \pi^{-1}\{p\}$ down to the base point $p$, we simply have that $\pi\circ (s_1 + fs_2)(p) = p$, and therefore we can conclude that $\pi \circ (s_1 + fs_2) \equiv Id_M$. For more details, check out chapter 10 of this book.