Identification of $T(T^*M)$ with $\mathbb{R}^{4n}$

Question

Given a smooth manifold $M^n$ consider the tangent bundle of the cotangent bundle $T(T^*M)$. It carries a natural structure of a smooth manifold of dimension $4n$. An element of $T(T^*M)$ can be identified as a pair $(y,v)$ with $y\in T^*M$ and $v\in T_yT^*M$. Hence, $y$ is itself can be idenetified with a pair $(p,u)$ with $p\in M$ and $u\in T^*_pM$. By definition of general tangent vectors, $v$ corresponds to an equivalence class of paths (for simplicity I will stick to a single such path) that satisfy $$\gamma:(-\delta,\delta)\to T^*M$$ and the derivatives in a local coordinate chart coincide. So $\gamma$ is of the form $\gamma(t)=(p(t),u(t))$ with $p(t)\in M$ and $u(t)\in T_{p(t)}^*M$ and $\gamma(0)=y$. This all means that we can view $v$ as a pair $v=(p'(0),u'(0))$ with $p'(0)\in T_pM$ and $u'(0)\in T_u(T_pM)$. (I think I did not mix the letters and notations and $u,p$ appear in correct places indicating just what they are suppose to). My question is how to make this rigorous? More precisely, $T(T^*M)$ is a manifold of dimension $4n$ and I would like to locally identify it with $\mathbb{R}^{4n}$ with coordinates as above. thanks for the help

Answer 1

Well you cannot identify the whole tangent/cotangent bundle with $\mathbb{R}^N$. This works only locally. Specifically, we have $T_x M \cong \mathbb{R}^n$ since every fiber of the tangent bundle of a $n$-dim manifold $M$ is some $n$-dim vector space. Since locally you have identification of neighbourhoods of $M$ with $\mathbb{R}^n$ via charts, you can identify locally $TM \cong \mathbb{R}^n \times \mathbb{R}^n$ and the coordinates can be taken such that one $\mathbb{R}^n$ is associated with some local coordinates on $M$ and the other $\mathbb{R}^n$ is associated with the coordinate vector fields of $T_xM$ (i.e. derivatives of the coordinate maps). Since the whole structure of tangent/cotangent bundle is a manifold, you can iterate this idea for the tangent bundle of this manifold, that is for $T T M$ (whether you consider tangent or cotangent bundles does not really make that much of a difference wrt our discussion).

Yet, as I read your post again, you are interested in a more specific description. Then I would suggest to pick a concrete chart on a given manifold and construct the chart on $TM$ and then do the same on $TTM$ following the idea of the above paragraph. Nevertheless, it seems to me that you already know how to do that.