Is it true that $A_p=\{\alpha(p): \alpha\in \Gamma(A)\}$ for a vector bundle $A\longrightarrow M$?

Question

Is it true that if $A\longrightarrow M$ is a vector bundle then $A_p=\{\alpha(p): \alpha\in \Gamma(A)\}$ for every $p\in M$?

Here $A_p$ is the fiber over $p$ and $\Gamma(A)$ are the smooth sections of $A$.

Thanks.

Answer 1

Yes, you can always extend any $\xi \in A_p$ to a section of $A$: Let $\phi: U \times \mathbb R^n \to A$ be a local trivialization near $p$ and $\psi : M \to \mathbb R$ be a smooth bump function with support compactly contained in $U$ and $\psi(p)=1$. Then we can define a section by $$\alpha(x) = \begin{cases}\psi(x) \phi(x,v) & \textrm{if }x \in U\\ 0 & \textrm{otherwise}\end{cases}$$

where $v \in \mathbb R^n$ are the components of $\xi$ in our trivialization; i.e. $\phi(p,v) =\xi$.