A smooth map $s\colon U\to E$ which is not a section where $\pi\colon E\to M$ is a vector bundle.

Question

I have a silly question about an example of a smooth map $s\colon U\to E$ which is not a section on $U$ where $\pi\colon E\to M$ is a vector bundle and $U$ is an open subspace of $M$. According to the definition, $s$ is going to be a section if it has a left inverse i.e. $$\pi\circ s=id_U.$$ So, if I take $M=U=\mathbb{R}$, $E=\mathbb{R}^2$, and $$s\colon\mathbb{R}\to \mathbb{R}^2\text{ with }s(t)=(\cos(t),\sin(t)),$$ then is it going to be an example of a smooth map that is not a section? What are the other examples? In other words, I want to understand the nature/motivation for sections.

Answer 1

A possible motivating example: take a smooth manifold $M$ and consider, for each $x\in M$ the tangent space $T_xM.$ Then, one defines the tangent bundle, $TM$ so that $TM$ is itself a smooth manifold. It is also a vector bundle. Its elements are pairs $(x,v)$ where $x\in M$ and $v\in T_xM.$ The intuition for this is that we gather together the individual vector spaces $T_xM$ and combine them into a differentiable manifold. Then, the projection $\pi:TM\to M$ sends $(x,v)$ to $x$, as expected, and the existence of a section $s$ ensures that when we send a particular $x\in M$ into $TM$, it ends up in $T_xM$ (considered as a subspace of $TM).$ That is, we want to exclude maps that send $x$ to elements of the form $(y,v): y\neq x.$ This requirement is built into the definition of section for if a map $s$ sends $x$ to such a $(y,v)$ then $\pi\circ s(x)=y\neq x.$