Condition for hypersurfaces in $mathbb{R}^n$ to be diffeomorphic?
$begingroup$
Let $n geq 2$ and consider two hypersurfaces $H_1$ and $H_2$ of $mathbb{R}^n$, as well as an application $phi: mathbb{R}^n to mathbb{R}^n$,such that the restriction of $phi$ from $H_1$ is bijective with values in $H_2$. What are in general the conditions required for the application $phi$ viewed as an application from $H_1$ to $H_2$ to be diffeomorphic? Do I need to check that the derivative of $phi$ does not cancel on the tangent space of any point of $H_1$?
Any reference on this kind of problem would be much appreciated.
geometry analysis derivatives differential-geometry
asked Jan 16 at 15:34
Gâteau-GalloisGâteau-Gallois
362112
$endgroup$
|
show 7 more comments
$begingroup$
Let $n geq 2$ and consider two hypersurfaces $H_1$ and $H_2$ of $mathbb{R}^n$, as well as an application $phi: mathbb{R}^n to mathbb{R}^n$,such that the restriction of $phi$ from $H_1$ is bijective with values in $H_2$. What are in general the conditions required for the application $phi$ viewed as an application from $H_1$ to $H_2$ to be diffeomorphic? Do I need to check that the derivative of $phi$ does not cancel on the tangent space of any point of $H_1$?
Any reference on this kind of problem would be much appreciated.
geometry analysis derivatives differential-geometry
asked Jan 16 at 15:34
Gâteau-GalloisGâteau-Gallois
362112
$endgroup$
$begingroup$
Are you asking what are the conditions such that the restricted map $phi : H_1 to H_2$ is a diffeomorphism? Or are you allowing that restriction to fail to be a diffeomorphisms, and still wanting to know whether $H_1$ and $H_2$ are diffeomorphic?
$endgroup$
– Lee Mosher
Jan 17 at 13:02
$begingroup$
The first one I believe.
$endgroup$
– Gâteau-Gallois
Jan 17 at 13:55
$begingroup$
Here is a sufficient condition for the restricted map $phi : H_1 to H_2$ to be a diffeomorphism: $phi$ itself is a diffeomorphism, and $H_1$ is a closed subset of $mathbb R^n$. Is that what you were looking for?
$endgroup$
– Lee Mosher
Jan 17 at 22:20
$begingroup$
Not really, I think this condition is not restrictive enough for what I want to do. For instance if I want to show that the surface of a convex set is diffeomorphic to the unit sphere, finding a map that is diffeomorphic from $mathbb{R}^n$ to itself would be quite hard I think.
$endgroup$
– Gâteau-Gallois
Jan 18 at 9:09
$begingroup$
But the surface of a convex set is not diffeomorphic to the unit sphere. Take the unit cube $[0,1]^n$, for example.
$endgroup$
– Lee Mosher
Jan 18 at 12:43
|
show 7 more comments
$begingroup$
Let $n geq 2$ and consider two hypersurfaces $H_1$ and $H_2$ of $mathbb{R}^n$, as well as an application $phi: mathbb{R}^n to mathbb{R}^n$,such that the restriction of $phi$ from $H_1$ is bijective with values in $H_2$. What are in general the conditions required for the application $phi$ viewed as an application from $H_1$ to $H_2$ to be diffeomorphic? Do I need to check that the derivative of $phi$ does not cancel on the tangent space of any point of $H_1$?
Any reference on this kind of problem would be much appreciated.
geometry analysis derivatives differential-geometry
asked Jan 16 at 15:34
Gâteau-GalloisGâteau-Gallois
362112
$endgroup$
Let $n geq 2$ and consider two hypersurfaces $H_1$ and $H_2$ of $mathbb{R}^n$, as well as an application $phi: mathbb{R}^n to mathbb{R}^n$,such that the restriction of $phi$ from $H_1$ is bijective with values in $H_2$. What are in general the conditions required for the application $phi$ viewed as an application from $H_1$ to $H_2$ to be diffeomorphic? Do I need to check that the derivative of $phi$ does not cancel on the tangent space of any point of $H_1$?
Any reference on this kind of problem would be much appreciated.
geometry analysis derivatives differential-geometry
geometry analysis derivatives differential-geometry
asked Jan 16 at 15:34
Gâteau-GalloisGâteau-Gallois
362112
asked Jan 16 at 15:34
Gâteau-GalloisGâteau-Gallois
362112
edited Jan 19 at 12:22
Gâteau-Gallois
asked Jan 16 at 15:34
Gâteau-GalloisGâteau-Gallois
362112
asked Jan 16 at 15:34
Gâteau-GalloisGâteau-Gallois
362112
asked Jan 16 at 15:34
Gâteau-GalloisGâteau-Gallois
362112
362112
$begingroup$
Are you asking what are the conditions such that the restricted map $phi : H_1 to H_2$ is a diffeomorphism? Or are you allowing that restriction to fail to be a diffeomorphisms, and still wanting to know whether $H_1$ and $H_2$ are diffeomorphic?
$endgroup$
– Lee Mosher
Jan 17 at 13:02
$begingroup$
The first one I believe.
$endgroup$
– Gâteau-Gallois
Jan 17 at 13:55
$begingroup$
Here is a sufficient condition for the restricted map $phi : H_1 to H_2$ to be a diffeomorphism: $phi$ itself is a diffeomorphism, and $H_1$ is a closed subset of $mathbb R^n$. Is that what you were looking for?
$endgroup$
– Lee Mosher
Jan 17 at 22:20
$begingroup$
Not really, I think this condition is not restrictive enough for what I want to do. For instance if I want to show that the surface of a convex set is diffeomorphic to the unit sphere, finding a map that is diffeomorphic from $mathbb{R}^n$ to itself would be quite hard I think.
$endgroup$
– Gâteau-Gallois
Jan 18 at 9:09
$begingroup$
But the surface of a convex set is not diffeomorphic to the unit sphere. Take the unit cube $[0,1]^n$, for example.
$endgroup$
– Lee Mosher
Jan 18 at 12:43
|
show 7 more comments
$begingroup$
Are you asking what are the conditions such that the restricted map $phi : H_1 to H_2$ is a diffeomorphism? Or are you allowing that restriction to fail to be a diffeomorphisms, and still wanting to know whether $H_1$ and $H_2$ are diffeomorphic?
$endgroup$
– Lee Mosher
Jan 17 at 13:02
$begingroup$
The first one I believe.
$endgroup$
– Gâteau-Gallois
Jan 17 at 13:55
$begingroup$
Here is a sufficient condition for the restricted map $phi : H_1 to H_2$ to be a diffeomorphism: $phi$ itself is a diffeomorphism, and $H_1$ is a closed subset of $mathbb R^n$. Is that what you were looking for?
$endgroup$
– Lee Mosher
Jan 17 at 22:20
$begingroup$
Not really, I think this condition is not restrictive enough for what I want to do. For instance if I want to show that the surface of a convex set is diffeomorphic to the unit sphere, finding a map that is diffeomorphic from $mathbb{R}^n$ to itself would be quite hard I think.
$endgroup$
– Gâteau-Gallois
Jan 18 at 9:09
$begingroup$
But the surface of a convex set is not diffeomorphic to the unit sphere. Take the unit cube $[0,1]^n$, for example.
$endgroup$
– Lee Mosher
Jan 18 at 12:43
$begingroup$
Are you asking what are the conditions such that the restricted map $phi : H_1 to H_2$ is a diffeomorphism? Or are you allowing that restriction to fail to be a diffeomorphisms, and still wanting to know whether $H_1$ and $H_2$ are diffeomorphic?
$endgroup$
– Lee Mosher
Jan 17 at 13:02
$begingroup$
Are you asking what are the conditions such that the restricted map $phi : H_1 to H_2$ is a diffeomorphism? Or are you allowing that restriction to fail to be a diffeomorphisms, and still wanting to know whether $H_1$ and $H_2$ are diffeomorphic?
$endgroup$
– Lee Mosher
Jan 17 at 13:02
$begingroup$
The first one I believe.
$endgroup$
– Gâteau-Gallois
Jan 17 at 13:55
$begingroup$
The first one I believe.
$endgroup$
– Gâteau-Gallois
Jan 17 at 13:55
$begingroup$
Here is a sufficient condition for the restricted map $phi : H_1 to H_2$ to be a diffeomorphism: $phi$ itself is a diffeomorphism, and $H_1$ is a closed subset of $mathbb R^n$. Is that what you were looking for?
$endgroup$
– Lee Mosher
Jan 17 at 22:20
$begingroup$
Here is a sufficient condition for the restricted map $phi : H_1 to H_2$ to be a diffeomorphism: $phi$ itself is a diffeomorphism, and $H_1$ is a closed subset of $mathbb R^n$. Is that what you were looking for?
$endgroup$
– Lee Mosher
Jan 17 at 22:20
$begingroup$
Not really, I think this condition is not restrictive enough for what I want to do. For instance if I want to show that the surface of a convex set is diffeomorphic to the unit sphere, finding a map that is diffeomorphic from $mathbb{R}^n$ to itself would be quite hard I think.
$endgroup$
– Gâteau-Gallois
Jan 18 at 9:09
$begingroup$
Not really, I think this condition is not restrictive enough for what I want to do. For instance if I want to show that the surface of a convex set is diffeomorphic to the unit sphere, finding a map that is diffeomorphic from $mathbb{R}^n$ to itself would be quite hard I think.
$endgroup$
– Gâteau-Gallois
Jan 18 at 9:09
$begingroup$
But the surface of a convex set is not diffeomorphic to the unit sphere. Take the unit cube $[0,1]^n$, for example.
$endgroup$
– Lee Mosher
Jan 18 at 12:43
$begingroup$
But the surface of a convex set is not diffeomorphic to the unit sphere. Take the unit cube $[0,1]^n$, for example.
$endgroup$
– Lee Mosher
Jan 18 at 12:43
|
show 7 more comments
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%2f3075877%2fcondition-for-hypersurfaces-in-mathbbrn-to-be-diffeomorphic%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%2f3075877%2fcondition-for-hypersurfaces-in-mathbbrn-to-be-diffeomorphic%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
$begingroup$
Are you asking what are the conditions such that the restricted map $phi : H_1 to H_2$ is a diffeomorphism? Or are you allowing that restriction to fail to be a diffeomorphisms, and still wanting to know whether $H_1$ and $H_2$ are diffeomorphic?
$endgroup$
– Lee Mosher
Jan 17 at 13:02
$begingroup$
The first one I believe.
$endgroup$
– Gâteau-Gallois
Jan 17 at 13:55
$begingroup$
Here is a sufficient condition for the restricted map $phi : H_1 to H_2$ to be a diffeomorphism: $phi$ itself is a diffeomorphism, and $H_1$ is a closed subset of $mathbb R^n$. Is that what you were looking for?
$endgroup$
– Lee Mosher
Jan 17 at 22:20
$begingroup$
Not really, I think this condition is not restrictive enough for what I want to do. For instance if I want to show that the surface of a convex set is diffeomorphic to the unit sphere, finding a map that is diffeomorphic from $mathbb{R}^n$ to itself would be quite hard I think.
$endgroup$
– Gâteau-Gallois
Jan 18 at 9:09
$begingroup$
But the surface of a convex set is not diffeomorphic to the unit sphere. Take the unit cube $[0,1]^n$, for example.
$endgroup$
– Lee Mosher
Jan 18 at 12:43