This is homework so no answers please
Here is the problem: Show that $UM:=\{(x,v)\in T\mathbb{R}^{n}:x\in M^{m}, v\in T_{x}M^{m},|v|=1\}$ is a (2m-1)-dim submanifold of $T\mathbb{R}^{n}$.
My attempt is ridiculously long, so I was wondering if I can get hints for a shorter one.
Also, is there a rigorous way to shrink open sets? At location (***), I need to shrink an open set $\pi^{-1}(U)$, to fit its image into another open set V. The problem is that both sets $\pi^{-1}(U)$, V are arbitrary and so I am not sure how to do rigorous shrinking.
Here is my attempt:
The goal is to show $UM$ satisfies the (2m-1) local slice criterion in $T\mathbb{R}^{n}$ for chart $(W,f)$. We will build f as a composition of charts.
$\blacktriangleright$ Since $M^{m}$ is an embedded submanifold, for each $p\in M$ there exists chart $(U,\phi)$ in $\mathbb{R}^{n}$ containing it s.t. $\phi(U\cap M)=\{(x_{1},...,x_{n})\in U: x_{m+1}=...=x_{n}=0\}$. The associated map for the tangent bundle is:
$\widetilde{\phi}:\pi^{-1}(U)\to \mathbb{R}^{2n}$ defined as $\widetilde{\phi}(v_{i}\frac{\partial }{\partial x_{i}}|_{p})=(x_{1},...,x_{n},v_{1},...,v_{n})$, where $\pi:T\mathbb{R}^{n}\to \mathbb{R}^{n}$.
Therefore, for $p\in M$ we have $\widetilde{\phi}(v_{i}\frac{\partial }{\partial x_{i}}|_{p})=(x_{1},...,x_{m},0,...,0,v_{1},...,v_{m},0,...,0)$.
$\blacktriangleright$ The idea is to compose $\widetilde{\phi}$ by diffeomorphic map $(v_{1},...,v_{m})\mapsto (v_{1},...,v_{m-1},0)$, where $|(v_{1},...,v_{m})|=1\Leftrightarrow (v_{1},...,v_{m})\in \mathbb{S}^{m-1}$, and so we get a chart on $\pi^{-1}(U)\cap UM$.
The $\mathbb{S}^{m-1}$ is an embedded (m-1)-dim submanifold of $\mathbb{R}^{m}$ and so we have some $(V,\psi)$ s.t. $\psi(V\cap \mathbb{S}^{m-1})=\{(x_{1},...,x_{m})\in V: x_{m}=0\}$. Then define $g:\mathbb{R}^{n}\times V\times \mathbb{R}^{n}\to \mathbb{R}^{n}$ as: $$g(x_{1},...,x_{n},v_{1},...,v_{n})=(x_{1},...,x_{n},\psi(v_{1},...,v_{m}),v_{m+1},...,v_{n})$$. The map g is a diffeomorphism as it is the identity in $\mathbb{R}^{n}\times \mathbb{R}^{n}$ and the diffeomorphic chart $\psi$ on V.
$\blacktriangleright$ We will show that the desired map is $f:=g\circ \widetilde{\phi}$ and $W:=\pi^{-1}(U)\cap UM$.\ Well-defined: g's domain is $\mathbb{R}^{n}\times V\times \mathbb{R}^{n}$, so we need to make sure $\pi_{(n+1,n+m)}(\widetilde{\phi}(\pi^{-1}(U) ))\subset V$, where $\pi_{(n+1,n+m)}:\mathbb{R}^{2n}\to \mathbb{R}^{m}$ be the projection onto $n+1,...,n+m$ coordinates. (***)
The map $g\circ \widetilde{\phi}:\pi^{-1}(U)\to \mathbb{R}^{2n}$ is a composition of diffeomorphisms and thus a coordinate chart on $\pi^{-1}(U)$. Now we need to show where it sents $UM\cap \pi^{-1}(U)$.
Derivations in $UM\cap \pi^{-1}(U)$ are tangent to M and so $\widetilde{\phi}(v_{i}\frac{\partial }{\partial x_{i}}|_{p})=(x_{1},...,x_{m},0,...,0,v_{1},...,v_{m},0,...,0)$. Since $(v_{1},...,v_{m})\in \mathbb{S}^{m-1}$, we get $\psi(v_{1},...,v_{m})=(v_{1},...,v_{m-1},0)$ and so $$g\circ \widetilde{\phi}(v_{i}\frac{\partial }{\partial x_{i}}|_{p})=g(x_{1},...,x_{m},0,...,0,v_{1},...,v_{m},0,...,0)=$$ $$=(x_{1},...,x_{m},0,...,0,v_{1},...,v_{m-1},0,0,...,0)$$.
Thus, $g\circ \widetilde{\phi}(UM\cap \pi^{-1}(U))=\{(x_{1},...,x_{n},v_{1},...,v_{n})\in g\circ \widetilde{\phi}(\pi^{-1}(U)): x_{m+1}=...=x_{n}=v_{m}=...=v_{n}=0\}$.
Thanks