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I'm not sure if I understood the smooth structure on tangent bundle of a smooth manifold, so my question is whether my understanding is correct, or not, so let me explain what I've understood so far.


Let $$TM = \cup_{p\in M} \{p\}\times T_p M,$$ and define $\sigma_p : T_pM \to \mathbb{R}^n$ by $\sigma_p ([c]) = D(\phi\circ c)(0)$, where $\phi$ is a local coordinate chart around $p$ and $c$ is a smooth path with $c(0) = p \in M$.Note that, the definition of the tangent space that I'm using given in the first paragraph of this question.

Let $U = dom(\phi)$, and $\sigma:R \subseteq TM \to \mathbb{R}^n \times \mathbb{R}^n $ be given by $$\sigma ((p, [c])) = (\phi(p), \sigma_p ([c])) =(\phi(p), D(\phi\circ c)(0)),$$ where $\pi_1 (R) = U.$

Now, we define a topology and a smooth structure using $\sigma$. One can easily check that, $\sigma$ is bijection, since when the first arguments are the same, $D(\phi\circ c_1)(0) = D(\phi\circ c_2)(0)$ implies $[c_1] = [c_2]$.Hence, we can put a topology on $TM$ in a way that $\sigma$ that makes $\sigma$ continuous by pulling the open sets in the product topology of $\mathbb{R}^n \times \mathbb{R}^n$. Since the coordinate charts are compatible, this in fact defines a unique topology on $TM$ that is independent of the charts chosen. To show that, observe the following:

Let $R := \sigma_\phi^{-1} (U \times V)$ for some open subsets $U,V\in \mathbb{R}^n$, i.e $R$ be an open subset of $TM$ wrt to the topology defined by using $\phi$. Then, observe that $$\sigma_\psi (R) = \sigma_\psi \circ \sigma_\phi^{-1} (U\times V) = ( \psi\circ \phi^{-1} (U), D(\psi \circ \phi^{-1})(V)).$$ Since this maps is a diffeomorphism in both coordinates, the image of $R$ under $\phi_\psi$ is open, hence the topology is independent of the coordinate charts. Moreover, this shows that we can cover $TM$ with $\sigma_\phi$'s in a compatible manner, hence such maps defines a natural smooth structure on this topological manifold, making into a smooth manifold.


Just to check: in the expression $$\sigma_\psi (R) = \sigma_\psi \circ \sigma_\phi^{-1} (U\times V) = ( \psi\circ \phi^{-1} (U), D(\psi \circ \phi^{-1})(V)),$$ is the second component correct ?

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