The developing map of conformally flat manifold
up vote
7
down vote
favorite
There is one sentence I don't understand in some paper.
"A simply connected and conformally flat three mainifold can be conformally immersed into $S^3$" by the means of a developing map.
Is any reference about this short argument? Maybe it is a direct consequence from definition. Could anyone explain a little bit to me?
dg.differential-geometry riemannian-geometry
asked Nov 28 at 5:04
H-H
1485
add a comment |
up vote
7
down vote
favorite
There is one sentence I don't understand in some paper.
"A simply connected and conformally flat three mainifold can be conformally immersed into $S^3$" by the means of a developing map.
Is any reference about this short argument? Maybe it is a direct consequence from definition. Could anyone explain a little bit to me?
dg.differential-geometry riemannian-geometry
asked Nov 28 at 5:04
H-H
1485
1
The general story is explained in Sharpe's book Differential Geometry: Cartan's Generalization of Klein's Erlangen Programme, but the argument is precisely Petrunin's below.
– Ben McKay
Nov 28 at 17:00
add a comment |
up vote
7
down vote
favorite
up vote
7
down vote
favorite
There is one sentence I don't understand in some paper.
"A simply connected and conformally flat three mainifold can be conformally immersed into $S^3$" by the means of a developing map.
Is any reference about this short argument? Maybe it is a direct consequence from definition. Could anyone explain a little bit to me?
dg.differential-geometry riemannian-geometry
asked Nov 28 at 5:04
H-H
1485
There is one sentence I don't understand in some paper.
"A simply connected and conformally flat three mainifold can be conformally immersed into $S^3$" by the means of a developing map.
Is any reference about this short argument? Maybe it is a direct consequence from definition. Could anyone explain a little bit to me?
dg.differential-geometry riemannian-geometry
dg.differential-geometry riemannian-geometry
asked Nov 28 at 5:04
H-H
1485
asked Nov 28 at 5:04
H-H
1485
asked Nov 28 at 5:04
H-H
1485
asked Nov 28 at 5:04
H-H
1485
asked Nov 28 at 5:04
H-H
1485
1485
1
The general story is explained in Sharpe's book Differential Geometry: Cartan's Generalization of Klein's Erlangen Programme, but the argument is precisely Petrunin's below.
– Ben McKay
Nov 28 at 17:00
add a comment |
1
The general story is explained in Sharpe's book Differential Geometry: Cartan's Generalization of Klein's Erlangen Programme, but the argument is precisely Petrunin's below.
– Ben McKay
Nov 28 at 17:00
1
1
The general story is explained in Sharpe's book Differential Geometry: Cartan's Generalization of Klein's Erlangen Programme, but the argument is precisely Petrunin's below.
– Ben McKay
Nov 28 at 17:00
The general story is explained in Sharpe's book Differential Geometry: Cartan's Generalization of Klein's Erlangen Programme, but the argument is precisely Petrunin's below.
– Ben McKay
Nov 28 at 17:00
add a comment |
1 Answer
1
active
oldest
votes
up vote
9
down vote
accepted
Since $mathbb{S}^3$ is conformally flat, we can think that any point of our manifold $M$ admits a neighborhood that is conformally equivalent to an open set in $mathbb{S}^3$.
If two such neighbohoods overlap then the corresponding gluing map between corresponding open sets in $mathbb{S}^3$ is a composition of inversions (Liouville's theorem).
So after applying a composition of inversions to one of the open sets you can assume that the gluing map is identity.
This way you can extend the parametrization of a neighborhood to an immersed neighborhood of any path.
The composition of inversions at a neighborhhod of the end point of path depends continuously on the path and therefore has to be the same for homotopic paths.
Since $M$ is simply connected it gives a well defined conformal immersion $Mhookrightarrow mathbb{S}^3$.
answered Nov 28 at 5:19
Anton Petrunin
26.6k580197
add a comment |
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
9
down vote
accepted
Since $mathbb{S}^3$ is conformally flat, we can think that any point of our manifold $M$ admits a neighborhood that is conformally equivalent to an open set in $mathbb{S}^3$.
If two such neighbohoods overlap then the corresponding gluing map between corresponding open sets in $mathbb{S}^3$ is a composition of inversions (Liouville's theorem).
So after applying a composition of inversions to one of the open sets you can assume that the gluing map is identity.
This way you can extend the parametrization of a neighborhood to an immersed neighborhood of any path.
The composition of inversions at a neighborhhod of the end point of path depends continuously on the path and therefore has to be the same for homotopic paths.
Since $M$ is simply connected it gives a well defined conformal immersion $Mhookrightarrow mathbb{S}^3$.
answered Nov 28 at 5:19
Anton Petrunin
26.6k580197
add a comment |
up vote
9
down vote
accepted
Since $mathbb{S}^3$ is conformally flat, we can think that any point of our manifold $M$ admits a neighborhood that is conformally equivalent to an open set in $mathbb{S}^3$.
If two such neighbohoods overlap then the corresponding gluing map between corresponding open sets in $mathbb{S}^3$ is a composition of inversions (Liouville's theorem).
So after applying a composition of inversions to one of the open sets you can assume that the gluing map is identity.
This way you can extend the parametrization of a neighborhood to an immersed neighborhood of any path.
The composition of inversions at a neighborhhod of the end point of path depends continuously on the path and therefore has to be the same for homotopic paths.
Since $M$ is simply connected it gives a well defined conformal immersion $Mhookrightarrow mathbb{S}^3$.
answered Nov 28 at 5:19
Anton Petrunin
26.6k580197
add a comment |
up vote
9
down vote
accepted
up vote
9
down vote
accepted
Since $mathbb{S}^3$ is conformally flat, we can think that any point of our manifold $M$ admits a neighborhood that is conformally equivalent to an open set in $mathbb{S}^3$.
If two such neighbohoods overlap then the corresponding gluing map between corresponding open sets in $mathbb{S}^3$ is a composition of inversions (Liouville's theorem).
So after applying a composition of inversions to one of the open sets you can assume that the gluing map is identity.
This way you can extend the parametrization of a neighborhood to an immersed neighborhood of any path.
The composition of inversions at a neighborhhod of the end point of path depends continuously on the path and therefore has to be the same for homotopic paths.
Since $M$ is simply connected it gives a well defined conformal immersion $Mhookrightarrow mathbb{S}^3$.
answered Nov 28 at 5:19
Anton Petrunin
26.6k580197
Since $mathbb{S}^3$ is conformally flat, we can think that any point of our manifold $M$ admits a neighborhood that is conformally equivalent to an open set in $mathbb{S}^3$.
If two such neighbohoods overlap then the corresponding gluing map between corresponding open sets in $mathbb{S}^3$ is a composition of inversions (Liouville's theorem).
So after applying a composition of inversions to one of the open sets you can assume that the gluing map is identity.
This way you can extend the parametrization of a neighborhood to an immersed neighborhood of any path.
The composition of inversions at a neighborhhod of the end point of path depends continuously on the path and therefore has to be the same for homotopic paths.
Since $M$ is simply connected it gives a well defined conformal immersion $Mhookrightarrow mathbb{S}^3$.
answered Nov 28 at 5:19
Anton Petrunin
26.6k580197
edited Nov 28 at 5:29
answered Nov 28 at 5:19
Anton Petrunin
26.6k580197
answered Nov 28 at 5:19
Anton Petrunin
26.6k580197
answered Nov 28 at 5:19
Anton Petrunin
26.6k580197
26.6k580197
add a comment |
add a comment |
Thanks for contributing an answer to MathOverflow!
- 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%2fmathoverflow.net%2fquestions%2f316379%2fthe-developing-map-of-conformally-flat-manifold%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
The general story is explained in Sharpe's book Differential Geometry: Cartan's Generalization of Klein's Erlangen Programme, but the argument is precisely Petrunin's below.
– Ben McKay
Nov 28 at 17:00