Integrable system is not a level set: an example
$begingroup$
For a manifold $M$, let $f in C^{k}(M), k>0$, be a level set (i.e. $rank(df)=1$). I understand that this trivially implies that ${{k.df: k in C^{k}(M)}}$ forms an integrable regular Pfaffian system, and that the level sets of $f$ form the integral manifolds of such a Pfaffian system.
The converse, I know, is not true. The best one has is the following: Given a Pfaffian system generated by a one-form $alpha$ that is integrable, there exists an open nbd. $U$ for every $m in M$ on which there exists $f, g in C^k(U)$ s.t. $g.alpha |_U=df$ where $g$ is nowhere zero, and $f^{-1}(t_0)$ corresponds to the integral manifolds restricted to $U$.
Yet I still see physics references talking about a global level set prescription for such a manifold $M$. Does someone know any simple example where the manifold has smooth foliations from integrability, but lacks a level set prescription?
differential-geometry pde differential-topology foliations
asked Jan 8 at 14:23
Sandesh JrSandesh Jr
315
$endgroup$
add a comment |
$begingroup$
For a manifold $M$, let $f in C^{k}(M), k>0$, be a level set (i.e. $rank(df)=1$). I understand that this trivially implies that ${{k.df: k in C^{k}(M)}}$ forms an integrable regular Pfaffian system, and that the level sets of $f$ form the integral manifolds of such a Pfaffian system.
The converse, I know, is not true. The best one has is the following: Given a Pfaffian system generated by a one-form $alpha$ that is integrable, there exists an open nbd. $U$ for every $m in M$ on which there exists $f, g in C^k(U)$ s.t. $g.alpha |_U=df$ where $g$ is nowhere zero, and $f^{-1}(t_0)$ corresponds to the integral manifolds restricted to $U$.
Yet I still see physics references talking about a global level set prescription for such a manifold $M$. Does someone know any simple example where the manifold has smooth foliations from integrability, but lacks a level set prescription?
differential-geometry pde differential-topology foliations
asked Jan 8 at 14:23
Sandesh JrSandesh Jr
315
$endgroup$
1
$begingroup$
You're requiring $fcolon MtoBbb R$. So take the foliation of $S^1times S^1$ by circles ${p}times S^1$. (You can do this with a $0$-dimensional foliation of $S^1$, of course, but I figured you'd prefer a higher-dimensional example.)
$endgroup$
– Ted Shifrin
Jan 8 at 17:23
$begingroup$
Ah, yes, of course!
$endgroup$
– Sandesh Jr
Jan 9 at 18:07
add a comment |
$begingroup$
For a manifold $M$, let $f in C^{k}(M), k>0$, be a level set (i.e. $rank(df)=1$). I understand that this trivially implies that ${{k.df: k in C^{k}(M)}}$ forms an integrable regular Pfaffian system, and that the level sets of $f$ form the integral manifolds of such a Pfaffian system.
The converse, I know, is not true. The best one has is the following: Given a Pfaffian system generated by a one-form $alpha$ that is integrable, there exists an open nbd. $U$ for every $m in M$ on which there exists $f, g in C^k(U)$ s.t. $g.alpha |_U=df$ where $g$ is nowhere zero, and $f^{-1}(t_0)$ corresponds to the integral manifolds restricted to $U$.
Yet I still see physics references talking about a global level set prescription for such a manifold $M$. Does someone know any simple example where the manifold has smooth foliations from integrability, but lacks a level set prescription?
differential-geometry pde differential-topology foliations
asked Jan 8 at 14:23
Sandesh JrSandesh Jr
315
$endgroup$
For a manifold $M$, let $f in C^{k}(M), k>0$, be a level set (i.e. $rank(df)=1$). I understand that this trivially implies that ${{k.df: k in C^{k}(M)}}$ forms an integrable regular Pfaffian system, and that the level sets of $f$ form the integral manifolds of such a Pfaffian system.
The converse, I know, is not true. The best one has is the following: Given a Pfaffian system generated by a one-form $alpha$ that is integrable, there exists an open nbd. $U$ for every $m in M$ on which there exists $f, g in C^k(U)$ s.t. $g.alpha |_U=df$ where $g$ is nowhere zero, and $f^{-1}(t_0)$ corresponds to the integral manifolds restricted to $U$.
Yet I still see physics references talking about a global level set prescription for such a manifold $M$. Does someone know any simple example where the manifold has smooth foliations from integrability, but lacks a level set prescription?
differential-geometry pde differential-topology foliations
differential-geometry pde differential-topology foliations
asked Jan 8 at 14:23
Sandesh JrSandesh Jr
315
asked Jan 8 at 14:23
Sandesh JrSandesh Jr
315
asked Jan 8 at 14:23
Sandesh JrSandesh Jr
315
asked Jan 8 at 14:23
Sandesh JrSandesh Jr
315
asked Jan 8 at 14:23
Sandesh JrSandesh Jr
315
315
1
$begingroup$
You're requiring $fcolon MtoBbb R$. So take the foliation of $S^1times S^1$ by circles ${p}times S^1$. (You can do this with a $0$-dimensional foliation of $S^1$, of course, but I figured you'd prefer a higher-dimensional example.)
$endgroup$
– Ted Shifrin
Jan 8 at 17:23
$begingroup$
Ah, yes, of course!
$endgroup$
– Sandesh Jr
Jan 9 at 18:07
add a comment |
1
$begingroup$
You're requiring $fcolon MtoBbb R$. So take the foliation of $S^1times S^1$ by circles ${p}times S^1$. (You can do this with a $0$-dimensional foliation of $S^1$, of course, but I figured you'd prefer a higher-dimensional example.)
$endgroup$
– Ted Shifrin
Jan 8 at 17:23
$begingroup$
Ah, yes, of course!
$endgroup$
– Sandesh Jr
Jan 9 at 18:07
1
1
$begingroup$
You're requiring $fcolon MtoBbb R$. So take the foliation of $S^1times S^1$ by circles ${p}times S^1$. (You can do this with a $0$-dimensional foliation of $S^1$, of course, but I figured you'd prefer a higher-dimensional example.)
$endgroup$
– Ted Shifrin
Jan 8 at 17:23
$begingroup$
You're requiring $fcolon MtoBbb R$. So take the foliation of $S^1times S^1$ by circles ${p}times S^1$. (You can do this with a $0$-dimensional foliation of $S^1$, of course, but I figured you'd prefer a higher-dimensional example.)
$endgroup$
– Ted Shifrin
Jan 8 at 17:23
$begingroup$
Ah, yes, of course!
$endgroup$
– Sandesh Jr
Jan 9 at 18:07
$begingroup$
Ah, yes, of course!
$endgroup$
– Sandesh Jr
Jan 9 at 18:07
add a comment |
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$begingroup$
You're requiring $fcolon MtoBbb R$. So take the foliation of $S^1times S^1$ by circles ${p}times S^1$. (You can do this with a $0$-dimensional foliation of $S^1$, of course, but I figured you'd prefer a higher-dimensional example.)
$endgroup$
– Ted Shifrin
Jan 8 at 17:23
$begingroup$
Ah, yes, of course!
$endgroup$
– Sandesh Jr
Jan 9 at 18:07