Definition of manifolds as submanifolds of $mathbb{R}^m$
up vote
2
down vote
favorite
I'm having trouble understanding the definition of coordinate charts of manifolds given in the book "Analysis II" by Herbert Amann and Joachim Escher. They define manifolds as submanifolds of $mathbb{R}^m$ (which is equivalent to the abstract aproach by the Whitney embedding theorem right?).
The definition of a submanifold I learned is:
Let $M$ be an $m$-dimensional manifold. We say that a subset $L subseteq M$ is a submanifold of $M$ of dimension $n$, if for every point $p in L$ there exists an adapted chart $phi: U rightarrow V' times V''$ with $U subseteq L$ open in $L$, $V' subseteq mathbb{R}^n$ open in $mathbb{R}^n$ and $V'' subseteq mathbb{R}^{m-n}$ open in $mathbb{R}^{m-n}$ such that $phi (U cap L) = V' times {0}$ with $0 in mathbb{R}^{m-n}$ Then $L$ becomes an $n$-dimensional manifold of itself with the smooth atlas induced by the restricted charts $phi : U cap L rightarrow V'$.
The authors of the book define a submanifold of $mathbb{R}^m$ as follows
A subset $L subseteq mathbb{R}^m$ is called an $n$-dimensional submanifold of $mathbb{R}^m$ if for every point $p in L$ there exists an open set $U subseteq mathbb{R}^m$ containing $p$ and an open subset $Vsubseteq mathbb{R}^m$ together with a diffeomorphism $phi$ from $U$ to $V$ such that $phi(M cap U)=V cap (mathbb{R}^n times {0})$ with $0 in mathbb{R}^{m-n}$.
,which directly coincides with the book's definition if one views $mathbb{R}^m$ as an $m$-dimensional manifold with the smooth atlas induced by the universal chart $(mathbb{R}^m, id)$.
Then the authors define coordinate charts of an $n$-dimensional submanifold $L$ of $mathbb{R}^m$ around a point $p in L$ as follows
Let $phi: U rightarrow V$ be a homeomorphism from an open set $U subseteq L$ in the subspace topology containing $p$ to an open subset $V$ of $mathbb{R}^n$ such that $ i_M circ phi ^{-1}$ is a $C^infty$-immersion, where $i_M$ is the cannonical injection from $L$ to $mathbb{R}^m$.
How is this equivalent to the restricted charts given in the first definition? Why does $ i_M circ phi^{-1}$ need to be an immersion? I'm really having trouble connecting the abstract aproach to manifolds to the way of defining manifolds as submanifolds of $mathbb{R}^m$. What's a good way to think about the approach taken by the book?
differential-geometry definition smooth-manifolds
asked Nov 22 at 20:03
Jannik Pitt
291316
add a comment |
up vote
2
down vote
favorite
I'm having trouble understanding the definition of coordinate charts of manifolds given in the book "Analysis II" by Herbert Amann and Joachim Escher. They define manifolds as submanifolds of $mathbb{R}^m$ (which is equivalent to the abstract aproach by the Whitney embedding theorem right?).
The definition of a submanifold I learned is:
Let $M$ be an $m$-dimensional manifold. We say that a subset $L subseteq M$ is a submanifold of $M$ of dimension $n$, if for every point $p in L$ there exists an adapted chart $phi: U rightarrow V' times V''$ with $U subseteq L$ open in $L$, $V' subseteq mathbb{R}^n$ open in $mathbb{R}^n$ and $V'' subseteq mathbb{R}^{m-n}$ open in $mathbb{R}^{m-n}$ such that $phi (U cap L) = V' times {0}$ with $0 in mathbb{R}^{m-n}$ Then $L$ becomes an $n$-dimensional manifold of itself with the smooth atlas induced by the restricted charts $phi : U cap L rightarrow V'$.
The authors of the book define a submanifold of $mathbb{R}^m$ as follows
A subset $L subseteq mathbb{R}^m$ is called an $n$-dimensional submanifold of $mathbb{R}^m$ if for every point $p in L$ there exists an open set $U subseteq mathbb{R}^m$ containing $p$ and an open subset $Vsubseteq mathbb{R}^m$ together with a diffeomorphism $phi$ from $U$ to $V$ such that $phi(M cap U)=V cap (mathbb{R}^n times {0})$ with $0 in mathbb{R}^{m-n}$.
,which directly coincides with the book's definition if one views $mathbb{R}^m$ as an $m$-dimensional manifold with the smooth atlas induced by the universal chart $(mathbb{R}^m, id)$.
Then the authors define coordinate charts of an $n$-dimensional submanifold $L$ of $mathbb{R}^m$ around a point $p in L$ as follows
Let $phi: U rightarrow V$ be a homeomorphism from an open set $U subseteq L$ in the subspace topology containing $p$ to an open subset $V$ of $mathbb{R}^n$ such that $ i_M circ phi ^{-1}$ is a $C^infty$-immersion, where $i_M$ is the cannonical injection from $L$ to $mathbb{R}^m$.
How is this equivalent to the restricted charts given in the first definition? Why does $ i_M circ phi^{-1}$ need to be an immersion? I'm really having trouble connecting the abstract aproach to manifolds to the way of defining manifolds as submanifolds of $mathbb{R}^m$. What's a good way to think about the approach taken by the book?
differential-geometry definition smooth-manifolds
asked Nov 22 at 20:03
Jannik Pitt
291316
add a comment |
up vote
2
down vote
favorite
up vote
2
down vote
favorite
I'm having trouble understanding the definition of coordinate charts of manifolds given in the book "Analysis II" by Herbert Amann and Joachim Escher. They define manifolds as submanifolds of $mathbb{R}^m$ (which is equivalent to the abstract aproach by the Whitney embedding theorem right?).
The definition of a submanifold I learned is:
Let $M$ be an $m$-dimensional manifold. We say that a subset $L subseteq M$ is a submanifold of $M$ of dimension $n$, if for every point $p in L$ there exists an adapted chart $phi: U rightarrow V' times V''$ with $U subseteq L$ open in $L$, $V' subseteq mathbb{R}^n$ open in $mathbb{R}^n$ and $V'' subseteq mathbb{R}^{m-n}$ open in $mathbb{R}^{m-n}$ such that $phi (U cap L) = V' times {0}$ with $0 in mathbb{R}^{m-n}$ Then $L$ becomes an $n$-dimensional manifold of itself with the smooth atlas induced by the restricted charts $phi : U cap L rightarrow V'$.
The authors of the book define a submanifold of $mathbb{R}^m$ as follows
A subset $L subseteq mathbb{R}^m$ is called an $n$-dimensional submanifold of $mathbb{R}^m$ if for every point $p in L$ there exists an open set $U subseteq mathbb{R}^m$ containing $p$ and an open subset $Vsubseteq mathbb{R}^m$ together with a diffeomorphism $phi$ from $U$ to $V$ such that $phi(M cap U)=V cap (mathbb{R}^n times {0})$ with $0 in mathbb{R}^{m-n}$.
,which directly coincides with the book's definition if one views $mathbb{R}^m$ as an $m$-dimensional manifold with the smooth atlas induced by the universal chart $(mathbb{R}^m, id)$.
Then the authors define coordinate charts of an $n$-dimensional submanifold $L$ of $mathbb{R}^m$ around a point $p in L$ as follows
Let $phi: U rightarrow V$ be a homeomorphism from an open set $U subseteq L$ in the subspace topology containing $p$ to an open subset $V$ of $mathbb{R}^n$ such that $ i_M circ phi ^{-1}$ is a $C^infty$-immersion, where $i_M$ is the cannonical injection from $L$ to $mathbb{R}^m$.
How is this equivalent to the restricted charts given in the first definition? Why does $ i_M circ phi^{-1}$ need to be an immersion? I'm really having trouble connecting the abstract aproach to manifolds to the way of defining manifolds as submanifolds of $mathbb{R}^m$. What's a good way to think about the approach taken by the book?
differential-geometry definition smooth-manifolds
asked Nov 22 at 20:03
Jannik Pitt
291316
I'm having trouble understanding the definition of coordinate charts of manifolds given in the book "Analysis II" by Herbert Amann and Joachim Escher. They define manifolds as submanifolds of $mathbb{R}^m$ (which is equivalent to the abstract aproach by the Whitney embedding theorem right?).
The definition of a submanifold I learned is:
Let $M$ be an $m$-dimensional manifold. We say that a subset $L subseteq M$ is a submanifold of $M$ of dimension $n$, if for every point $p in L$ there exists an adapted chart $phi: U rightarrow V' times V''$ with $U subseteq L$ open in $L$, $V' subseteq mathbb{R}^n$ open in $mathbb{R}^n$ and $V'' subseteq mathbb{R}^{m-n}$ open in $mathbb{R}^{m-n}$ such that $phi (U cap L) = V' times {0}$ with $0 in mathbb{R}^{m-n}$ Then $L$ becomes an $n$-dimensional manifold of itself with the smooth atlas induced by the restricted charts $phi : U cap L rightarrow V'$.
The authors of the book define a submanifold of $mathbb{R}^m$ as follows
A subset $L subseteq mathbb{R}^m$ is called an $n$-dimensional submanifold of $mathbb{R}^m$ if for every point $p in L$ there exists an open set $U subseteq mathbb{R}^m$ containing $p$ and an open subset $Vsubseteq mathbb{R}^m$ together with a diffeomorphism $phi$ from $U$ to $V$ such that $phi(M cap U)=V cap (mathbb{R}^n times {0})$ with $0 in mathbb{R}^{m-n}$.
,which directly coincides with the book's definition if one views $mathbb{R}^m$ as an $m$-dimensional manifold with the smooth atlas induced by the universal chart $(mathbb{R}^m, id)$.
Then the authors define coordinate charts of an $n$-dimensional submanifold $L$ of $mathbb{R}^m$ around a point $p in L$ as follows
Let $phi: U rightarrow V$ be a homeomorphism from an open set $U subseteq L$ in the subspace topology containing $p$ to an open subset $V$ of $mathbb{R}^n$ such that $ i_M circ phi ^{-1}$ is a $C^infty$-immersion, where $i_M$ is the cannonical injection from $L$ to $mathbb{R}^m$.
How is this equivalent to the restricted charts given in the first definition? Why does $ i_M circ phi^{-1}$ need to be an immersion? I'm really having trouble connecting the abstract aproach to manifolds to the way of defining manifolds as submanifolds of $mathbb{R}^m$. What's a good way to think about the approach taken by the book?
differential-geometry definition smooth-manifolds
differential-geometry definition smooth-manifolds
asked Nov 22 at 20:03
Jannik Pitt
291316
asked Nov 22 at 20:03
Jannik Pitt
291316
asked Nov 22 at 20:03
Jannik Pitt
291316
asked Nov 22 at 20:03
Jannik Pitt
291316
asked Nov 22 at 20:03
Jannik Pitt
291316
291316
add a comment |
add a comment |
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',
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%2f3009584%2fdefinition-of-manifolds-as-submanifolds-of-mathbbrm%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
active
oldest
votes
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.
Some of your past answers have not been well-received, and you're in danger of being blocked from answering.
Please pay close attention to the following guidance:
- 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.
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%2f3009584%2fdefinition-of-manifolds-as-submanifolds-of-mathbbrm%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
