3
$\begingroup$

I'm trying to answer problem 4-3 from Lee's Introduction to Smooth Manifolds, 2nd edition. The problem says:

Formulate and prove a version of the rank theorem for a map of constant rank whose domain is a smooth manifold with boundary.

Lee himself gave a hint in another question. Here is what I have so far:

I'm assuming I have a smooth map of constant rank $F: \mathbb{H}^m\rightarrow\mathbb{R}^n$. Let's say $\mathrm{rank}(F)=k$. Then I extend $F$ to a smooth map $\tilde{F}:\mathbb{R}^m\rightarrow\mathbb{R}^n$. I can shrink the domain to a small enough neighborhood of $\mathbf{0}$, $U$, such that there is a projection $\pi:\mathbb{R}^n\rightarrow\mathbb{R}^k$ with $\pi\tilde{F}_{|U}$ a submersion. The actual constant-rank theorem then says we have charts $(A,\alpha)$ and $(B,\beta)$ with $$ \beta\circ\pi\tilde{F}\circ\alpha^{-1}:\alpha(A)\rightarrow\beta(B)$$

If I write the coordinates of $\mathbb{H}^m$ as $(x,y)$ and the coordinates of of $\alpha(A)$ as $(a,t)$, then this map looks like $$ \beta\circ\pi\tilde{F}\circ\alpha^{-1}(a,t) = a$$

I can switch the order so the projection is the last map applied: $$ \pi\circ(\beta\times\mathrm{id})\circ\tilde{F}\circ\alpha^{-1}\equiv\beta\circ\pi\tilde{F}\circ\alpha^{-1}$$

Which finally means $$(\beta\times\mathrm{id})\circ\tilde{F}\circ\alpha^{-1}(a,t)=(a,S(a,t))$$ Where $S:\alpha(A)\rightarrow\mathbb{R}^{(n-k)}$.

This is pretty similar to how the constant-rank theorem is proved, but now I'm stuck. The map above, restricted to $\alpha(A\cap\mathbb{H}^m)$, has rank $k$. But that does not mean $S$ is independent of $y$. That's because $\alpha$ is not necessarily a boundary chart on $A\cap\mathbb{H}^m$. I also don't see any way to make $\alpha$ a boundary chart (something similar is proved in Lee's book, but crucially uses that $F$ is an immersion).

Can someone give me a hint as to how to finish this? Ideally I'd get to a place where (with possibly different charts) $$ \tilde{\beta}\circ F\circ\tilde{\alpha}^{-1}(a,t) = (a,0)$$ and $\tilde{\alpha}$ is a boundary chart.

Bonus question: where do I use Lee's assumption (from the hint) that $\ker dF_p\not\subseteq T_p\partial\mathbb{H}^m$?

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.