I'm learning a bit about smooth manifolds, and currently I'm learning about tangent bundles (just definitions mainly) and vector field.
This is my reference : Tu's Introduction to Manifolds.
I was also watching a video about tangent bundles (because I was struggling with the concept).
Once I understood the definition however I realized I have an issue understanding why we need the notion of tangent bundle. I can't remember where I read this but am I right when I say that tangent bundles are necessary if we want to generalize the notion of function on a manifold?
Consider a smooth manifold $M$, if we wanted to define what a vector field is to me the definition should reflect the fact that for each $p \in M$ we have a $v \in T_p M$, therefore it should be a map.
This is probably the key why such association isn't good as definition because a map needs both domain (in this case $M$) and an image space, however my naive definition involves for each $p$ a different space $T_p M$ and this is why we need the notion of tangent bundle.
Is this observation correct?
Best Answer
I think the simplest motivation is that of vector fields. We want to be able to assign a tangent vector to each point in the manifold, giving us a "field" of vectors on the manifold. That is, $F$ should be some map such that
$$ F(p)\in T_pM $$
for $p\in M$. So, what's wrong with just saying that? Well, nothing really, if you're only interested in the value of vector fields at a point. If you ever want to look beyond singular points, you need some structure connecting your different tangent spaces. For example, to look at continuity of $F$, we need the space of outputs of $F$ to have a topological structure; to look at differentiability, we need a differentiable structure.
So, we need some space which
To satisfy (1), we simply glue together all the tangent spaces by taking a union. Since the tangent spaces are completely disconnected from one another, we can exemplify this fact by using a disjoint union
$$ TM = \coprod\limits_{p\in M} T_pM $$
The rest of the bundle structure is just there to "lift" the structure of $M$ onto $TM$. With that, we can now define vector fields as functions in the usual way
$$ F: M\to TM $$
and we are able to discuss continuity and differentiability to the extent that $M$ admits such properties. However, this definition isn't "complete" because it allows for e.g. attaching a tangent vector from $T_qM$ to $p$, which doesn't fit our idea of a vector field. This gives us another requirement,
This is the bundle projection map, $\pi: TM\to M$, and so we can add the requirement on $F$ that $\pi(F(p)) = p$ everywhere.
Here, we created a bundle with a base manifold and a vector space at each point, but we could imagine a more general concept of bundle which just has some kind of space $B$ with some other kind of space $F_p B$ at each point $p\in B$. Even in this general setting, we can see the utility of attaching to each point of the base space some element of its attached space. We define a function
$$ \sigma: B\to FB = E $$
with $\pi(\sigma(p)) = p$ as a cross-section (or just section) of the total space $E$.
Applying this terminology to our original example, we can then reconstruct the more terse definition: