Topology and smooth structure on tangent bundle

Question

My lecture notes on differential geometry read the following (without proof):

For $M$ a manifold, let $TM = \bigcup_{p \in M} T_p M$ be the (disjoint) union of all its tangent spaces. Then, there exists a unique topology and smooth structure on $TM$ making it into a smooth manifold such that any section $X:M \rightarrow TM$ of the canonical projection map is smooth if and only if for all smooth functions $f$ on $M$, the function $Xf$ is smooth.

I am not quite sure as to how to approach this (I am talking about the uniqueness part, the usual topology and smooth structure evidently imply the desired equivalence). I found a related post without an answer here: Why is the manifold structure on the tangent bundle unique? It seems that in this post, OP also assumes the projection map to be continuous, whereas my lecture notes do not.

Any help would be appreciated.

Answer 1

This is two separate results tied in one. I'm not going to write all the details here since they're quite long and besides, their complete proof were all written in great details in Lee's $\textit{Introduction to Smooth Manifolds, 2nd ed}$ . The first one, $TM$ as smooth manifold, is proved in $\textbf{Proposition 3.18.}$, and the uniqueness follows from a lemma used to proved the proposition (Lemma 1.35).

The second is about characterization of smooth vector fields $X : M \to TM$, proved in $\textbf{Proposition 8.14}$., says that a vector field $$ X : M \to TM \text{ is smooth } \Leftrightarrow \forall f \in C^{\infty}(M), \text{ the function }Xf \text{ is smooth on }M \Leftrightarrow \text{ For every open subset } U\subseteq M \text{ and } \forall f \in C^{\infty}(U) \text{ then function } Xf \text{ is smooth on } U. $$ Good Luck.

Answer 2

Since it seems that awarding the bounty isn't going to do much, I'll post my own (partial) answer anyways:

By Proposition 10.24 in Introduction to Smooth Manifolds (Lee), we can deduce the following:

Theorem: The natural topology and smooth structure on the set $TM = \cup_{p \in M} T_p M$ are unique when we require the following:

(i) $\pi: TM \rightarrow M$ is a smooth vector bundle, where $\pi$ is the canonical projection map $(p,v_p) \mapsto p$.

(ii) For $U \subset M$ open and $X: U \rightarrow TM$ an arbitrary local section, if $Xf \in C^{\infty}(U)$ for all $f \in C^{\infty}(U)$ then $X$ is smooth.

Proof: Indeed, it follows from property (ii) that local coordinate vector fields are smooth, so that we can apply Proposition 10.24.

Note that we did require the manifold structure of $TM$ to be of 'bundle' type. Intuitively, it would seem that we want the zero section to be a smooth embedding of $M$ into $TM$. Then, knowing the topology on the fibers as well and requiring a local 'product structure' seems to determine everything. As I pointed out in one of my comments above, it is not likely that we can achieve any such result without smoothness requirements on some map which goes out of $TM$, and the only canonical one we have is the projection. Perhaps condition (i) can be weakened to something like '$\pi$ is a smooth submersion'?