Exercise 3.19 (Tangent bundle of a smooth manifold with boundary is a smooth manifold with boundary)

Question

Suppose $M$ is a smooth manifold with boundary. Show that $TM$ has a natural topology and smooth structure making it into a smooth manifold with boundary, such that if $\left(U,\left(x^i\right)\right)$ is any smooth boundary chart for $M$, then rearranging the coordinates in the natural chart $\left(\pi^{-1}(U),\left(x^i,v^i\right)\right)$ for $TM$ yields a boundary chart $\left(\pi^{-1}(U),\left(v^i,x^i\right)\right)$.

Elu Thingol · Accepted Answer

We use Smooth Manifold Chart Lemma (\href{./exercise_1-43}{Exerise 1.43}) the same way that Lemma 1.35 was used in Proposition 3.18. We must, however, modify (3.13) and add an additional argument to account for the boundary charts.
ewline

Let $\mathcal{A} = \{(U_{\alpha},\varphi_{\alpha})\}_{\alpha \in A}$ be the smooth structure of the manifold with boundary $M$. For each coordinate map $(U_{\alpha},\varphi_{\alpha})$ in $\mathcal{A}$ define
\begin{equation}
    \widetilde{\varphi}_{\alpha}: 
    \underbrace{\pi^{-1}(U_{\alpha})}_{\subseteq TM} 
    	o 
    \underbrace{	extcolor{blue}{\mathbb{R}^n} 	imes 	extcolor{red}{\varphi(U_{\alpha})}}_{\subseteq \mathbb{R}^n 	imes \mathbb{H}^{n} = \mathbb{H}^{2n}}, \quad
    \widetilde{\varphi}_{\alpha}\begin{pmatrix}	extcolor{red}{p}, 	extcolor{blue}{\displaystyle\sum_{i=1}^{n} v^i \left.\dfrac{\partial}{\partial x^i}ight|_p}\end{pmatrix} 
    =
    \begin{pmatrix}
        	extcolor{blue}{\displaystyle\sum_{i=1}^{n} v^i e^i}, 	extcolor{red}{\varphi_{\alpha}(p)}
    \end{pmatrix}.
    	ag{3.13}
\end{equation}
In other words, we have a collection $\{\pi^{-1}(U_\alpha)\}_{\alpha \in A}$ of subsets of $M$ together with maps $\widetilde{\varphi}_{\alpha}: \pi^{-1}(U_\alpha) 	o \mathbb{H}^{2n}$. The rest of the proof follows analogously to the proof of Proposition 3.18.

Exercise 3.19 (Tangent bundle of a smooth manifold with boundary is a smooth manifold with boundary)

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Exercise 3.19 (Tangent bundle of a smooth manifold with boundary is a smooth manifold with boundary)

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