I'm reading John Lee's Introduction to smooth manifolds.
In problem 5-6, He asked to show that for embedded submanifold $M^{n}$ of $\mathbb{R}^{m}$, $UM=\{(x,v)\in T\mathbb{R}^m \; |\; |v|=1\}$ is $2n-1$ dimensional embedded submanifold of $T\mathbb{R}^m$ where metric is induced from $T\mathbb{R}^m \sim\mathbb{R}^{2m}$.
I first thought this also holds for the case of $\partial M \not= \varnothing$ and also believed that I proved in that case too, using half slices.
But in problme 6-2, He asked to show (weak) Whitney immersion theorem using $G:UM^{2n-1}\to \mathbb{RP}^{2n}$ and Sard's theorem when $\partial M\not=\varnothing$ with remark to see problem 9-14 for general case.
This implies to me that problem 5-6 doesn't work for $\partial M\not=\varnothing$.
Does my guess correct? If so, please give me some counter example or points where proof of problem 5-6 doesn't work in $\partial M\not=\varnothing$ case.
Thank you.
The result of Problem 5-6 is certainly true when $\partial M\ne \varnothing$ (with "submanifold" replaced by "submanifold with boundary"). But what I had in mind for that problem was to use the regular level set theorem (Corollary 5.14) on $TM$, and that theorem is not true as stated on a manifold with boundary. (It's true with some additional hypotheses, which do hold in this case. But I didn't want to spend time going into those extra hypotheses. The argument for Whitney immersion using the double seems more economical and conceptually simple.)