I'm reading through Tu's Introduction to manifolds and today I learned about tangent bundles and vector bundles. I was surprised to learn that $TS^2$, tangent bundle of the 2-sphere, isn't trivial (i.e. $TS^2 \not \simeq S^2 \times \mathbb R^2$. I learned that it could be seen as a corollary to the Hairy ball theorem.
I know the question is extremely vague, but how does $TS^2$ look like then? What do we know about it's topological/differential properties? Is there some way we can visualize it?
What are the other ways to see the non-triviality of $TS^2$?
I know I'm little out of my depth here and that I probably won't understand all the answers, but I hope they could motivate me to learn more of the differential geometry. Also, it's always good to get a little taste of what's ahead before you see your first definition.
As a corollary of the hairy ball theorem, $S^2$ is not a parallelizable manifold, that is, it does not have a set of $2$ globally non-vanishing vector fields that span its tangent space at every point. The parallelizable vector fields can be used to introduce a trivialization.