Structure of double tangent bundle

I have a problem in understanding the structure of the tangent bundle of the tangent bundle of a manifold.

Let $M$ be an n-dimensional smooth manifold. Let $(U,\varphi)$ be a chart of $M$, i.e. $U\subset M$ open and $\varphi:U\rightarrow\varphi(U)\subset\mathbb{R}^n$ is an homeomorphism. Assume that $\varphi$ induces local coordinates $(x^1,...,x^n)$, that means that $\forall p\in U$, $\varphi(p)=(x^1(p),...,x^n(p))\in\mathbb{R}^n$. Let $p\in U$, then since $\varphi:U\subset M\rightarrow\varphi(U)\subset\mathbb{R}^n$ is an homeomorphism, then $(\varphi^{-1})_*|_{\varphi(p)}$ is an isomorphism of vector spaces: \begin{equation*} (\varphi^{-1})_*|_{\varphi(p)}:T_{\varphi(p)}\varphi(U)\cong T_{\varphi(p)}\mathbb{R}^n\rightarrow T_pU\cong T_pM. \end{equation*} Let \begin{equation} \label{basis of TR} \left\{\frac{\partial}{\partial x^1}|_{\varphi(p)},...\frac{\partial}{\partial x^n}|_{\varphi(p)}\right\} \end{equation} be a basis of $T_{\varphi(p)}\mathbb{R}^n$, then define \begin{equation*} \frac{\partial}{\partial x^i}|_{p}:=(\varphi^{-1})_*|_{\varphi(p)}\left(\frac{\partial}{\partial x^i}|_{\varphi(p)}\right). \end{equation*} By the fact that $(\varphi^{-1})_*|_{\varphi(p)}$ is an isomorphism, it maps basis into basis, hence \begin{equation*} \left\{\frac{\partial}{\partial x^1}|_{p},...,\frac{\partial}{\partial x^n}|_{p}\right\} \end{equation*} is a basis of $T_pM$. In other words \begin{equation*} T_pM=\left\{V^i\frac{\partial}{\partial x^i}|_{p}\mid V^i\in\mathbb{R}, \ i=1,...,n\right\}. \end{equation*} This means that any $V\in T_pM$, there exist $\{V^i\}_{i=1}^n\subset\mathbb{R}^n$ such that $V=V^i\frac{\partial}{\partial x^i}|_{p}$.

Consider now a smooth vector field V of $M$, i.e. $V\in\Gamma(TM)$ is a section of the tangent bundle $TM:=\coprod_{p\in M}T_pM$, then $\forall p\in U$, $V(p)\in T_pM$, hence we can write $V(p)=V^i(p)\frac{\partial}{\partial x^i}|_{p}$, for some $V^i(p)\in\mathbb{R}$, for $i=1,...,n$. Since this can be done $\forall p\in U$, we write locally \begin{equation*} V=V^i\frac{\partial}{\partial x^i}, \end{equation*} where $V^i\in\mathcal{C}^\infty(M)$ are suitable smooth functions. The word locally means that $V(p)=V^i(p)\frac{\partial}{\partial x^i}|_{p}$ only for $p\in U$.

Since $TM$ is a manifold by its own right we would like to define its charts. We can induce a charts of $TM$ by charts of $M$ in this way. Let $\pi:TM\rightarrow M$ be the projection of the bundle and take $V\in TM$, with $\pi(V)=p\in U\subset M$, then we have shown that we can write $V=V^i\frac{\partial}{\partial x^i}|_p$. Then we define the chart $(\tilde{U},\psi)$, where $\tilde{U}:=\pi^{-1}(U)\subset TM$ and \begin{equation} \label{chart TM} \begin{array}{rcl} \psi:\tilde{U}&\rightarrow&\psi(\tilde{U})\subset\mathbb{R}^{2n}\\ V=V^i\frac{\partial}{\partial x^i}|_p&\mapsto&(\varphi(p);V^1,...,V^n) \end{array} \end{equation}

Since $TM$ is a manifold we can consider its tangent bundle $\pi_{TM}:TTM\rightarrow TM$. Let $(x^1,...,x^n,y^1,...,y^n)$ be the local coordinates induced by $\psi$. Let $V\in \tilde{U}$, then since $\psi:\tilde{U}\subset TM\rightarrow\psi(\tilde{U})\subset\mathbb{R}^{2n}$ is an homeomorphism, then $(\psi^{-1})_*|_{\psi(V)}$ is an isomorphism of vector spaces: \begin{equation*} (\psi^{-1})_*|_{\psi(V)}:T_{\psi(V)}\psi(\tilde{U})\cong T_{\psi(V)}\mathbb{R}^{2n}\rightarrow T_VT\tilde{U}\cong T_VTM. \end{equation*} Let \begin{equation} \label{basis of TTR} \left\{\frac{\partial}{\partial x^1}|_{\psi(V)},...\frac{\partial}{\partial x^n}|_{\psi(V)};\frac{\partial}{\partial y^1}|_{\psi(V)},...\frac{\partial}{\partial y^n}|_{\psi(V)}\right\} \end{equation} be a basis of $T_{\psi(V)}\mathbb{R}^{2n}$, then define for $i=1,...,n$ \begin{equation*} \frac{\partial}{\partial x^i}|_{V}:=(\psi^{-1})_*|_{\psi(V)}\left(\frac{\partial}{\partial x^i}|_{\psi(V)}\right) \end{equation*} and \begin{equation*} \frac{\partial}{\partial y^i}|_{V}:=(\psi^{-1})_*|_{\psi(V)}\left(\frac{\partial}{\partial y^i}|_{\psi(V)}\right). \end{equation*} By the fact that $(\psi^{-1})_*|_{\psi(V)}$ is an isomorphism, it maps basis into basis, hence \begin{equation*} \left\{\frac{\partial}{\partial x^1}|_{V},...,\frac{\partial}{\partial x^n}|_{V};\frac{\partial}{\partial y^1}|_{V},...,\frac{\partial}{\partial y^n}|_{V}\right\} \end{equation*} is a basis of $T_VTM$. In other words \begin{equation*} T_VTM=\left\{\eta^i\frac{\partial}{\partial x^i}|_{V}+\tilde{\eta}^j\frac{\partial}{\partial y^j}|_{V}\mid \eta^i,\tilde{\eta}^j\in\mathbb{R}, \ i,j=1,...,n\right\}. \end{equation*}

Wikipedia instead says (https://en.wikipedia.org/wiki/Double_tangent_bundle):

Since $(TM,\pi_{TM},M)$ is a vector bundle on its own right, its tangent bundle has the secondary vector bundle structure $(TTM,(\pi_{TM})_*,TM)$, where $(\pi_{TM})_*:TTM\rightarrow TM$ is the push-forward of the canonical projection $\pi_{TM}:TM\rightarrow M$. In the following we denote \begin{equation} \xi = \xi^k\frac{\partial}{\partial x^k}\Big|_x\in T_xM, \qquad X = X^k\frac{\partial}{\partial x^k}\Big|_x\in T_xM \end{equation}

and apply the associated coordinate system

\begin{equation} \xi \mapsto (x^1,\ldots,x^n,\xi^1,\ldots,\xi^n) \end{equation}

on $TM$. Then the fibre of the secondary vector bundle structure at $X\in T_xM$ takes the form

\begin{equation} (\pi_{TM})^{-1}_*(X) = \Big\{ \ X^k\frac{\partial}{\partial x^k}\Big|_\xi + Y^k\frac{\partial}{\partial\xi^k}\Big|_\xi \ \Big| \ \xi\in T_xM \ , \ Y^1,\ldots,Y^n\in\mathbb{R} \ \Big\}. \end{equation}.

Well, this is different from my fibre \begin{equation*} T_VTM=\left\{\eta^i\frac{\partial}{\partial x^i}|_{V}+\tilde{\eta}^j\frac{\partial}{\partial y^j}|_{V}\mid \eta^i,\tilde{\eta}^j\in\mathbb{R}, \ i,j=1,...,n\right\}. \end{equation*}

What is the right fibre? Maybe they are just different vector bundle with the same base space $TM$, so what is their difference? Thank you everyone.

Solution 1:

Here is where you went wrong. A point in $TM$ is not just a vector from $T_pM$; instead, it's a pair $(p,v)$ where $p \in M$ and $v \in T_pM$. Therefore $\pi: TM \to M$ is not injective and so does not have an inverse.