Intersection of two plaques is an open subset of both plaques.
$begingroup$
I'm following the book Foliations I, and on pages 20 and 21, where they first define a foliation on a manifold, it is claimed that, implicit from the definition is the fact that when two plaques from distinct foliated charts intersect, the intersection is open in both plaques. I've been having a hard time proving this using the definition given in the book.
(With another definition of a foliation via a maximal foliated chart where the change of coordinates have a nice behavior this is easy, but for now I'm trying use the first definition. In any case this book proves both definitions are equivalent, but it uses the result I'm having trouble with in a lemma to do so).
Let $P$ and $Q$ be intersecting plaques associated to foliated charts $(U,varphi)$ and $(V,psi),$ respectively, of a $q$-codimentional foliation of an $n$-manifold $M$, with $P=varphi^{-1}(varphi(U)cap(mathbb R^{n-q}times {a}))$ and $Q = psi^{-1}(psi(V)cap(mathbb R^{n-q}times {b})),$ for some $a,b in mathbb R^q.$ I tried to compute the images of $Pcap Q$ from the two charts, but I led me nowhere. I can't see any other way to do it.
Am I missing something here? They make it sound very obvious but I can't really see it. Thanks in advance.
differential-geometry manifolds smooth-manifolds foliations
asked Dec 22 '18 at 6:43
VicVic
427
$endgroup$
add a comment |
$begingroup$
I'm following the book Foliations I, and on pages 20 and 21, where they first define a foliation on a manifold, it is claimed that, implicit from the definition is the fact that when two plaques from distinct foliated charts intersect, the intersection is open in both plaques. I've been having a hard time proving this using the definition given in the book.
(With another definition of a foliation via a maximal foliated chart where the change of coordinates have a nice behavior this is easy, but for now I'm trying use the first definition. In any case this book proves both definitions are equivalent, but it uses the result I'm having trouble with in a lemma to do so).
Let $P$ and $Q$ be intersecting plaques associated to foliated charts $(U,varphi)$ and $(V,psi),$ respectively, of a $q$-codimentional foliation of an $n$-manifold $M$, with $P=varphi^{-1}(varphi(U)cap(mathbb R^{n-q}times {a}))$ and $Q = psi^{-1}(psi(V)cap(mathbb R^{n-q}times {b})),$ for some $a,b in mathbb R^q.$ I tried to compute the images of $Pcap Q$ from the two charts, but I led me nowhere. I can't see any other way to do it.
Am I missing something here? They make it sound very obvious but I can't really see it. Thanks in advance.
differential-geometry manifolds smooth-manifolds foliations
asked Dec 22 '18 at 6:43
VicVic
427
$endgroup$
add a comment |
$begingroup$
I'm following the book Foliations I, and on pages 20 and 21, where they first define a foliation on a manifold, it is claimed that, implicit from the definition is the fact that when two plaques from distinct foliated charts intersect, the intersection is open in both plaques. I've been having a hard time proving this using the definition given in the book.
(With another definition of a foliation via a maximal foliated chart where the change of coordinates have a nice behavior this is easy, but for now I'm trying use the first definition. In any case this book proves both definitions are equivalent, but it uses the result I'm having trouble with in a lemma to do so).
Let $P$ and $Q$ be intersecting plaques associated to foliated charts $(U,varphi)$ and $(V,psi),$ respectively, of a $q$-codimentional foliation of an $n$-manifold $M$, with $P=varphi^{-1}(varphi(U)cap(mathbb R^{n-q}times {a}))$ and $Q = psi^{-1}(psi(V)cap(mathbb R^{n-q}times {b})),$ for some $a,b in mathbb R^q.$ I tried to compute the images of $Pcap Q$ from the two charts, but I led me nowhere. I can't see any other way to do it.
Am I missing something here? They make it sound very obvious but I can't really see it. Thanks in advance.
differential-geometry manifolds smooth-manifolds foliations
asked Dec 22 '18 at 6:43
VicVic
427
$endgroup$
I'm following the book Foliations I, and on pages 20 and 21, where they first define a foliation on a manifold, it is claimed that, implicit from the definition is the fact that when two plaques from distinct foliated charts intersect, the intersection is open in both plaques. I've been having a hard time proving this using the definition given in the book.
(With another definition of a foliation via a maximal foliated chart where the change of coordinates have a nice behavior this is easy, but for now I'm trying use the first definition. In any case this book proves both definitions are equivalent, but it uses the result I'm having trouble with in a lemma to do so).
Let $P$ and $Q$ be intersecting plaques associated to foliated charts $(U,varphi)$ and $(V,psi),$ respectively, of a $q$-codimentional foliation of an $n$-manifold $M$, with $P=varphi^{-1}(varphi(U)cap(mathbb R^{n-q}times {a}))$ and $Q = psi^{-1}(psi(V)cap(mathbb R^{n-q}times {b})),$ for some $a,b in mathbb R^q.$ I tried to compute the images of $Pcap Q$ from the two charts, but I led me nowhere. I can't see any other way to do it.
Am I missing something here? They make it sound very obvious but I can't really see it. Thanks in advance.
differential-geometry manifolds smooth-manifolds foliations
differential-geometry manifolds smooth-manifolds foliations
asked Dec 22 '18 at 6:43
VicVic
427
asked Dec 22 '18 at 6:43
VicVic
427
edited Dec 23 '18 at 5:58
Vic
asked Dec 22 '18 at 6:43
VicVic
427
asked Dec 22 '18 at 6:43
VicVic
427
asked Dec 22 '18 at 6:43
VicVic
427
427
add a comment |
add a comment |
0
active
oldest
votes
Your Answer
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%2f3049173%2fintersection-of-two-plaques-is-an-open-subset-of-both-plaques%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%2f3049173%2fintersection-of-two-plaques-is-an-open-subset-of-both-plaques%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
