The tangent bundle as a smooth manifold.

A reference for this material is Chapter 3 of John M. Lee. Introduction to smooth manifolds. Second edition. Grad. Texts in Math., 218. Springer, New York, 2013. xvi+708pp. ISBN: 978-1-4419-9981-8.

\(TM\) has a natural structure of smooth manifold. We now present an atlas. Let \(\pi : TM \to M \) be the function with \(\pi (T_pM) = \{ p \}\) for all \(p \in M\). For each chart \((U, \varphi ) \) of \(M\), one can produce the chart \((\pi ^{-1} (U) , \tilde {\varphi })\) of \(TM\) with

\[ \tilde {\varphi } \left ( \sum _{i = 1 }^n a_i \, \partial _i \vert _x \right ) : = ( \varphi (x) , a_1, \ldots , a_n ) . \]

We now check that the change of coordinates is smooth. To do that, we need to be more specific with our notation. Pick two charts \((U,\varphi )\), \((V, \psi )\) of \(M\). We need to show that \(\tilde {\psi } \circ \tilde {\varphi } ^{-1}\) is smooth.

On \(\varphi (U) \) we use coordinates \((x^1, \ldots , x^n)\), and on \(\psi (V) \) we use coordinates \((y^1 , \ldots , y^n)\). By the chain rule, for each \(j \in \{ 1,\ldots , n \} \) we have

\[ \frac {\partial \,\,\, }{\partial x^j} = \sum _{ i = 1 } ^n \frac {\partial y^i}{\partial x ^j} \frac {\partial \,\, }{\partial y^i} . \]

Therefore, whenever defined,

\begin{align*} \tilde {\psi } \circ \tilde {\varphi } ^{-1} (x , a_1, \ldots , a_n) & = \tilde {\psi } \left ( \sum _{ j = 1 } ^n a_j \frac {\partial }{\partial x^j } \Big \vert _{ \varphi ^{-1} (x) } \right ) \\ & = \tilde {\psi } \left ( \sum _{ j = 1 } ^n \sum _{i = 1 } ^n a_j \frac {\partial y^i}{\partial x ^j} \frac {\partial \,\, }{\partial y^i} \Big \vert _{ \varphi ^{-1} (x) } \right ) \\ & = \tilde {\psi } \left ( \sum _{ i = 1 } ^n \sum _{j = 1 } ^n a_j \frac {\partial y^i}{\partial x ^j} \frac {\partial \,\, }{\partial y^i} \Big \vert _{ \varphi ^{-1} (x) } \right ) \\ & = \left ( \psi ( \varphi ^{-1} (x)) , \sum _{j = 1 } ^n a_j \frac {\partial y^1}{\partial x ^j} , \ldots , \sum _{j = 1 } ^n a_j \frac {\partial y^n}{\partial x ^j} \right ) . \end{align*}

This expression depends smoothly on \(x\), \(a_1\), \(\ldots \), \(a_n\), so the change of coordinates is smooth. Therefore, the charts of the form \((\pi ^{-1} (U) , \tilde {\varphi })\) define a smooth structure on \(TM\).