Constant Rank Theorem for Manifolds with Boundary
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 $F: mathbb{H}^mrightarrowmathbb{R}^n$. Let's say $mathrm{rank}(F)=k$. Then I extend $F$ to a smooth map $tilde{F}:mathbb{R}^mrightarrowmathbb{R}^n$. I can shrink the domain to a small enough neighborhood of $mathbf{0}$, $U$, such that there is a projection $pi:mathbb{R}^nrightarrowmathbb{R}^k$ with $pitilde{F}_{|U}$ a submersion. The actual constant-rank theorem then says we have charts $(A,alpha)$ and $(B,beta)$ with
$$ betacircpitilde{F}circalpha^{-1}:alpha(A)rightarrowbeta(B)$$
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
$$ betacircpitilde{F}circalpha^{-1}(a,t) = a$$
I can switch the order so the projection is the last map applied:
$$ picirc(betatimesmathrm{id})circtilde{F}circalpha^{-1}equivbetacircpitilde{F}circalpha^{-1}$$
Which finally means
$$(betatimesmathrm{id})circtilde{F}circalpha^{-1}(a,t)=(a,S(a,t))$$
Where $S:alpha(A)rightarrowmathbb{R}^{(n-k)}$.
This is pretty similar to how the constant-rank theorem is proved, but now I'm stuck. The map above, restricted to $alpha(Acapmathbb{H}^m)$, has rank $k$. But that does not mean $S$ is independent of $y$. That's because $alpha$ is not necessarily a boundary chart on $Acapmathbb{H}^m$. 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)
$$ tilde{beta}circ Fcirctilde{alpha}^{-1}(a,t) = (a,0)$$
and $tilde{alpha}$ is a boundary chart.
Bonus question: where do I use Lee's assumption (from the hint) that $ker dF_pnotsubseteq T_ppartialmathbb{H}^m$?
differential-geometry smooth-manifolds manifolds-with-boundary smooth-functions
asked Nov 27 at 0:41
Hempelicious
122110
add a comment |
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 $F: mathbb{H}^mrightarrowmathbb{R}^n$. Let's say $mathrm{rank}(F)=k$. Then I extend $F$ to a smooth map $tilde{F}:mathbb{R}^mrightarrowmathbb{R}^n$. I can shrink the domain to a small enough neighborhood of $mathbf{0}$, $U$, such that there is a projection $pi:mathbb{R}^nrightarrowmathbb{R}^k$ with $pitilde{F}_{|U}$ a submersion. The actual constant-rank theorem then says we have charts $(A,alpha)$ and $(B,beta)$ with
$$ betacircpitilde{F}circalpha^{-1}:alpha(A)rightarrowbeta(B)$$
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
$$ betacircpitilde{F}circalpha^{-1}(a,t) = a$$
I can switch the order so the projection is the last map applied:
$$ picirc(betatimesmathrm{id})circtilde{F}circalpha^{-1}equivbetacircpitilde{F}circalpha^{-1}$$
Which finally means
$$(betatimesmathrm{id})circtilde{F}circalpha^{-1}(a,t)=(a,S(a,t))$$
Where $S:alpha(A)rightarrowmathbb{R}^{(n-k)}$.
This is pretty similar to how the constant-rank theorem is proved, but now I'm stuck. The map above, restricted to $alpha(Acapmathbb{H}^m)$, has rank $k$. But that does not mean $S$ is independent of $y$. That's because $alpha$ is not necessarily a boundary chart on $Acapmathbb{H}^m$. 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)
$$ tilde{beta}circ Fcirctilde{alpha}^{-1}(a,t) = (a,0)$$
and $tilde{alpha}$ is a boundary chart.
Bonus question: where do I use Lee's assumption (from the hint) that $ker dF_pnotsubseteq T_ppartialmathbb{H}^m$?
differential-geometry smooth-manifolds manifolds-with-boundary smooth-functions
asked Nov 27 at 0:41
Hempelicious
122110
add a comment |
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 $F: mathbb{H}^mrightarrowmathbb{R}^n$. Let's say $mathrm{rank}(F)=k$. Then I extend $F$ to a smooth map $tilde{F}:mathbb{R}^mrightarrowmathbb{R}^n$. I can shrink the domain to a small enough neighborhood of $mathbf{0}$, $U$, such that there is a projection $pi:mathbb{R}^nrightarrowmathbb{R}^k$ with $pitilde{F}_{|U}$ a submersion. The actual constant-rank theorem then says we have charts $(A,alpha)$ and $(B,beta)$ with
$$ betacircpitilde{F}circalpha^{-1}:alpha(A)rightarrowbeta(B)$$
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
$$ betacircpitilde{F}circalpha^{-1}(a,t) = a$$
I can switch the order so the projection is the last map applied:
$$ picirc(betatimesmathrm{id})circtilde{F}circalpha^{-1}equivbetacircpitilde{F}circalpha^{-1}$$
Which finally means
$$(betatimesmathrm{id})circtilde{F}circalpha^{-1}(a,t)=(a,S(a,t))$$
Where $S:alpha(A)rightarrowmathbb{R}^{(n-k)}$.
This is pretty similar to how the constant-rank theorem is proved, but now I'm stuck. The map above, restricted to $alpha(Acapmathbb{H}^m)$, has rank $k$. But that does not mean $S$ is independent of $y$. That's because $alpha$ is not necessarily a boundary chart on $Acapmathbb{H}^m$. 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)
$$ tilde{beta}circ Fcirctilde{alpha}^{-1}(a,t) = (a,0)$$
and $tilde{alpha}$ is a boundary chart.
Bonus question: where do I use Lee's assumption (from the hint) that $ker dF_pnotsubseteq T_ppartialmathbb{H}^m$?
differential-geometry smooth-manifolds manifolds-with-boundary smooth-functions
asked Nov 27 at 0:41
Hempelicious
122110
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 $F: mathbb{H}^mrightarrowmathbb{R}^n$. Let's say $mathrm{rank}(F)=k$. Then I extend $F$ to a smooth map $tilde{F}:mathbb{R}^mrightarrowmathbb{R}^n$. I can shrink the domain to a small enough neighborhood of $mathbf{0}$, $U$, such that there is a projection $pi:mathbb{R}^nrightarrowmathbb{R}^k$ with $pitilde{F}_{|U}$ a submersion. The actual constant-rank theorem then says we have charts $(A,alpha)$ and $(B,beta)$ with
$$ betacircpitilde{F}circalpha^{-1}:alpha(A)rightarrowbeta(B)$$
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
$$ betacircpitilde{F}circalpha^{-1}(a,t) = a$$
I can switch the order so the projection is the last map applied:
$$ picirc(betatimesmathrm{id})circtilde{F}circalpha^{-1}equivbetacircpitilde{F}circalpha^{-1}$$
Which finally means
$$(betatimesmathrm{id})circtilde{F}circalpha^{-1}(a,t)=(a,S(a,t))$$
Where $S:alpha(A)rightarrowmathbb{R}^{(n-k)}$.
This is pretty similar to how the constant-rank theorem is proved, but now I'm stuck. The map above, restricted to $alpha(Acapmathbb{H}^m)$, has rank $k$. But that does not mean $S$ is independent of $y$. That's because $alpha$ is not necessarily a boundary chart on $Acapmathbb{H}^m$. 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)
$$ tilde{beta}circ Fcirctilde{alpha}^{-1}(a,t) = (a,0)$$
and $tilde{alpha}$ is a boundary chart.
Bonus question: where do I use Lee's assumption (from the hint) that $ker dF_pnotsubseteq T_ppartialmathbb{H}^m$?
differential-geometry smooth-manifolds manifolds-with-boundary smooth-functions
differential-geometry smooth-manifolds manifolds-with-boundary smooth-functions
asked Nov 27 at 0:41
Hempelicious
122110
asked Nov 27 at 0:41
Hempelicious
122110
asked Nov 27 at 0:41
Hempelicious
122110
asked Nov 27 at 0:41
Hempelicious
122110
asked Nov 27 at 0:41
Hempelicious
122110
122110
add a comment |
add a comment |
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