I am trying to prove the following proposition:
Proposition: Let M be a smooth manifold and let $\pi:TM\to M$ be the projection to M, i.e, $\pi(p,v_p)=p$.
Then p is smooth and a submersion.
I found the following proof in another thread that the natural projection is smooth:
Let $\pi:T(X)\to X$, be the projection map in question.
Let $U\subset T(X)$ be a neighborhood of some $(x',v')\in T(X)$ and $V\subset X$ a neighborhood of $\pi(x',v')=x'\in X$. We know that there exists a chart $\psi:V\to \psi(V)\subset \mathbb R^k$.
Let $\phi:U\to \phi(U)\in\mathbb R^{2k}$ be a local coordinate chart for $T(X)$ where for any $(x,v)\in U$, $\phi((x,v))=(x_1,\ldots,x_k,\ldots, x_{2k})$ where $(x_1,\ldots,x_k)=\psi(x)$ for some $x\in V$. In that case, notice that the following diagram commutes $\newcommand{\ra}[1]{\kern-1.5ex\xrightarrow{\ \ #1\ \ }\phantom{}\kern-1.5ex} \newcommand{\ras}[1]{\kern-1.5ex\xrightarrow{\ \ \smash{#1}\ \ }\phantom{}\kern-1.5ex} \newcommand{\da}[1]{\bigg\downarrow\raise.5ex\rlap{\scriptstyle#1}}$ \begin{array}{c} U & \ra{\pi|_U} & V \\ \da{\phi} & & \da{\psi} \\ \phi(U) & \ras{\nu} & \psi(V) \\ \end{array} where $\nu:\phi(U)\to\psi(V)$ is the canonical submersion i.e, $$\nu(x_1,\ldots,x_k,\ldots, x_{2k})=(x_1,\ldots,x_k)=\psi(x)$$ Clearly, $\psi\circ \pi|_U\circ \phi^{-1}=\nu$. Thus by the local submersion theorem, $\pi$ is a submersion at $(x',v')$. Since our proof was independent of the choice of $(x',v')$, we conclude that $\pi$ is a submersion everywhere in $T(X)$
However I am not understanding how the author applies the local submersion theorem. I think the author is making the reverse implication that is if $v$ is the canonical projection then $\pi$ is a submersion. But the theorem states the other way around. The version I am familiar with is the following: Local Submersion theorem: Consider $\mathbb{R}^n=\mathbb{R}^{m_1}\times\mathbb{R}^{m_2}$ and let $U\subseteq\mathbb{R}^n$ open with $p\in U$. Let $f:U\to\mathbb{R}^{m_2}$ such that $D_2 f(p)$ is a linear isomorphism. Then, there is an open neighborhood $V_p\subseteq U$ of $p$, and an open set $w_p$ of $\mathbb{R}^n$ and a diffeomorphism $\phi:w_p\to V_p$ such that $f\circ\phi=\pi_2$.
Questions:
How do I finish this proof? How do I apply the local submersion theorem?
Thanks in advance!
