The difference between a fiber and a section of a vector bundle

Question

If $ E_x := \pi^{-1}(x) $ is the fiber over $x$ where $(E,\pi,M)$ is the vector bundle. And the section is $s: M \to E $ with $\pi \circ s = id_M $. This implies that $\pi^{-1} = s $ on $M$. So then whats the difference between a fiber over $x$ and the section restricted to $x$.

Thanks.

Answer 1

A section is a choice of a particular element from each fiber. That is, $s(x) \in E_x$. Thus if we have a section $s$ and we restrict it to a point $x$, then we get a single vector from the fiber $E_x$. On the other hand, the fiber $E_x$ is generally more than a single vector (it's a vector space).

Answer 2

A fibre is a set and a section is a map. The fibre over $x$ is the set $\pi^{-1}(x)$ and the section is the map $s:M\to E$

Answer 3

A section is any function s that assigns to every point p in the base an element in its fiber $E_p$ The restriction s(p) assigns to p a point in $E_p$. As an example, a section of the tangent bundle is a vector field, i.e., an assignment of a tangent vector to each tangent space $T_pM$; its restriction to p assigns to p a tangent vector $X_p \in T_pM$ . The fiber over a point $p$ is the inverse image of $p$ under the specific map $\pi$. In a vector bundle, this fiber is a vector space. In the same example of the tangent bundle to a manifold $M$ , with $\pi(x,p)=p$ , $\pi^{-1}(p)=T_pM$.