Concrete example of zero section

Question

I just learnt tangent bundle and I want to get some intuition about zero section (and sections in general). I'm even not clear about what the zero vector is in a tangent space--e.g. just consider $df_p:T_p\mathbb{R}=\mathbb{R}\to T_{f(p)}\mathbb{R}=\mathbb{R}$, is $0\in\mathbb{R}$ the zero vector in $T_{f(p)}\mathbb{R}$ (which gives a critical point)? Can someone explain a good concrete example of zero section to me?

Answer 1

Recall the general definition of a section. Let $\pi:E\to B$ be a (vector) bundle. A section is a map $s:B\to E$, such that $\pi\circ s=id$. In other words, a section is a map $B\to E$ that carries any point in $B$ to some point in the fiber over it.

A section of the tangent bundle is just a vector field.

Assume now $E\to B$ is a vector bundle. Then the fiber over a point $p\in B$ is a vector space, and it has a zero element $0_p$. The zero section is the map $s:B\to E$ that carries any point $p$ to $0_p$.

Edit: If $E$ is the tangent bundle, the zero section can be thought of in two ways, depending on your idea of tangent vectors. If you think of tangent vectors as equivalence classes of paths, then the zero tangent vector $0_p$ is the equivalence class represented by the constant path $\gamma\equiv p$. If you think of tangent vectors as derivations, then $0_p$ is the zero derivation, which maps every function to $0$.