Mixing
Conformal group of closed ball
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I read recently the follow affirmation on book "The Smith Conjectura" written by John W. Morgan and Hyman Bass : Every conformal transformation of unit sphere $S^2$ is has an unique conformal extention to closed unit ball , and this conformal extention is an isometry of hiperbolic space $H^{n + 1}$ as ball model ". My question is if this affirmation is true in dimention $n geq 2$. Of more way exact, I want of know if the relation between conformal transformation group is true. I read some texts, o Liouville Theorem, but not find nothings about the closed case.
riemannian-geometry
asked Jan 10 at 17:56
A.D.A.D.
375110
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add a comment |
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I read recently the follow affirmation on book "The Smith Conjectura" written by John W. Morgan and Hyman Bass : Every conformal transformation of unit sphere $S^2$ is has an unique conformal extention to closed unit ball , and this conformal extention is an isometry of hiperbolic space $H^{n + 1}$ as ball model ". My question is if this affirmation is true in dimention $n geq 2$. Of more way exact, I want of know if the relation between conformal transformation group is true. I read some texts, o Liouville Theorem, but not find nothings about the closed case.
riemannian-geometry
asked Jan 10 at 17:56
A.D.A.D.
375110
$endgroup$
1
$begingroup$
This is a direct corollary of Liouville's theorem: Every conformal automorphism of $S^n$ is Moebius and Moebius transformations are exactly the restrictions of isometries of the hyperbolic space $H^{n+1}$ in the unit ball model. One way to see this is that each Moebius transformation is a composition of inversions and each inversion acting on $S^n$ extends uniquely to an inversion acting on $S^{n+1}$.
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– Moishe Cohen
Jan 11 at 0:14
add a comment |
$begingroup$
I read recently the follow affirmation on book "The Smith Conjectura" written by John W. Morgan and Hyman Bass : Every conformal transformation of unit sphere $S^2$ is has an unique conformal extention to closed unit ball , and this conformal extention is an isometry of hiperbolic space $H^{n + 1}$ as ball model ". My question is if this affirmation is true in dimention $n geq 2$. Of more way exact, I want of know if the relation between conformal transformation group is true. I read some texts, o Liouville Theorem, but not find nothings about the closed case.
riemannian-geometry
asked Jan 10 at 17:56
A.D.A.D.
375110
$endgroup$
I read recently the follow affirmation on book "The Smith Conjectura" written by John W. Morgan and Hyman Bass : Every conformal transformation of unit sphere $S^2$ is has an unique conformal extention to closed unit ball , and this conformal extention is an isometry of hiperbolic space $H^{n + 1}$ as ball model ". My question is if this affirmation is true in dimention $n geq 2$. Of more way exact, I want of know if the relation between conformal transformation group is true. I read some texts, o Liouville Theorem, but not find nothings about the closed case.
riemannian-geometry
riemannian-geometry
asked Jan 10 at 17:56
A.D.A.D.
375110
asked Jan 10 at 17:56
A.D.A.D.
375110
asked Jan 10 at 17:56
A.D.A.D.
375110
asked Jan 10 at 17:56
A.D.A.D.
375110
asked Jan 10 at 17:56
A.D.A.D.
375110
375110
1
$begingroup$
This is a direct corollary of Liouville's theorem: Every conformal automorphism of $S^n$ is Moebius and Moebius transformations are exactly the restrictions of isometries of the hyperbolic space $H^{n+1}$ in the unit ball model. One way to see this is that each Moebius transformation is a composition of inversions and each inversion acting on $S^n$ extends uniquely to an inversion acting on $S^{n+1}$.
$endgroup$
– Moishe Cohen
Jan 11 at 0:14
add a comment |
1
$begingroup$
This is a direct corollary of Liouville's theorem: Every conformal automorphism of $S^n$ is Moebius and Moebius transformations are exactly the restrictions of isometries of the hyperbolic space $H^{n+1}$ in the unit ball model. One way to see this is that each Moebius transformation is a composition of inversions and each inversion acting on $S^n$ extends uniquely to an inversion acting on $S^{n+1}$.
$endgroup$
– Moishe Cohen
Jan 11 at 0:14
1
1
$begingroup$
This is a direct corollary of Liouville's theorem: Every conformal automorphism of $S^n$ is Moebius and Moebius transformations are exactly the restrictions of isometries of the hyperbolic space $H^{n+1}$ in the unit ball model. One way to see this is that each Moebius transformation is a composition of inversions and each inversion acting on $S^n$ extends uniquely to an inversion acting on $S^{n+1}$.
$endgroup$
– Moishe Cohen
Jan 11 at 0:14
$begingroup$
This is a direct corollary of Liouville's theorem: Every conformal automorphism of $S^n$ is Moebius and Moebius transformations are exactly the restrictions of isometries of the hyperbolic space $H^{n+1}$ in the unit ball model. One way to see this is that each Moebius transformation is a composition of inversions and each inversion acting on $S^n$ extends uniquely to an inversion acting on $S^{n+1}$.
$endgroup$
– Moishe Cohen
Jan 11 at 0:14
add a comment |
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This is a direct corollary of Liouville's theorem: Every conformal automorphism of $S^n$ is Moebius and Moebius transformations are exactly the restrictions of isometries of the hyperbolic space $H^{n+1}$ in the unit ball model. One way to see this is that each Moebius transformation is a composition of inversions and each inversion acting on $S^n$ extends uniquely to an inversion acting on $S^{n+1}$.
$endgroup$
– Moishe Cohen
Jan 11 at 0:14