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
1218
$endgroup$
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
1218
$endgroup$
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
1218
$endgroup$
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
1218
asked Jan 8 at 14:23
Sandesh JrSandesh Jr
1218
asked Jan 8 at 14:23
Sandesh JrSandesh Jr
1218
asked Jan 8 at 14:23
Sandesh JrSandesh Jr
1218
asked Jan 8 at 14:23
Sandesh JrSandesh Jr
1218
1218
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 |
0
active
oldest
votes
StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");
StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});
function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3066227%2fintegrable-system-is-not-a-level-set-an-example%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
0
active
oldest
votes
0
active
oldest
votes
active
oldest
votes
active
oldest
votes
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3066227%2fintegrable-system-is-not-a-level-set-an-example%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
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