Mixing
Conformal homeomorphism
0
$begingroup$
I was reading this paper
https://arxiv.org/pdf/math/0106036.pdf
and it talks about conformal homeomorphism, can someone give me the definition please?
complex-geometry
asked Jan 31 at 22:39
Claudio DelfinoClaudio Delfino
63
$endgroup$
|
show 1 more comment
0
$begingroup$
I was reading this paper
https://arxiv.org/pdf/math/0106036.pdf
and it talks about conformal homeomorphism, can someone give me the definition please?
complex-geometry
asked Jan 31 at 22:39
Claudio DelfinoClaudio Delfino
63
$endgroup$
$begingroup$
How much Riemannian geometry do you know?
$endgroup$
– Blazej
Jan 31 at 23:25
$begingroup$
I know what is an homeomorphism and what is a conformal map, but i never faced the conformal homeomorphism
$endgroup$
– Claudio Delfino
Feb 1 at 1:39
$begingroup$
What do would you say a conformal map is?
$endgroup$
– user98602
Feb 1 at 16:32
$begingroup$
For me a conformal map is an $fcolon D to D'$ , where $D,D' subset mathbb{C}$, $f$ analytic and one to one and onto (which should implies that $f'(z)ne0$ for all z and $f^{-1}$ is a conformal map too)
$endgroup$
– Claudio Delfino
Feb 1 at 17:31
$begingroup$
What you refer as a conformal map is a special case. If you have two Riemannian manifolds $M, N$ with metrics $g$ and $h$, conformal homeomorphism $M to N$ would be a diffeomorphism $phi : M to N$ such that pullback of $h$ coincides with $g$ multiplied by some function. If $M$ and $N$ are open subsets of complex plane this is equivalent to the condition you quoted (provided that you restrict attention to orientation preserving maps; otherwise you have to include anti-holomorphic functions also).
$endgroup$
– Blazej
Feb 1 at 21:37
|
show 1 more comment
0
0
0
$begingroup$
I was reading this paper
https://arxiv.org/pdf/math/0106036.pdf
and it talks about conformal homeomorphism, can someone give me the definition please?
complex-geometry
asked Jan 31 at 22:39
Claudio DelfinoClaudio Delfino
63
$endgroup$
I was reading this paper
https://arxiv.org/pdf/math/0106036.pdf
and it talks about conformal homeomorphism, can someone give me the definition please?
complex-geometry
complex-geometry
asked Jan 31 at 22:39
Claudio DelfinoClaudio Delfino
63
asked Jan 31 at 22:39
Claudio DelfinoClaudio Delfino
63
asked Jan 31 at 22:39
Claudio DelfinoClaudio Delfino
63
asked Jan 31 at 22:39
Claudio DelfinoClaudio Delfino
63
asked Jan 31 at 22:39
Claudio DelfinoClaudio Delfino
63
63
$begingroup$
How much Riemannian geometry do you know?
$endgroup$
– Blazej
Jan 31 at 23:25
$begingroup$
I know what is an homeomorphism and what is a conformal map, but i never faced the conformal homeomorphism
$endgroup$
– Claudio Delfino
Feb 1 at 1:39
$begingroup$
What do would you say a conformal map is?
$endgroup$
– user98602
Feb 1 at 16:32
$begingroup$
For me a conformal map is an $fcolon D to D'$ , where $D,D' subset mathbb{C}$, $f$ analytic and one to one and onto (which should implies that $f'(z)ne0$ for all z and $f^{-1}$ is a conformal map too)
$endgroup$
– Claudio Delfino
Feb 1 at 17:31
$begingroup$
What you refer as a conformal map is a special case. If you have two Riemannian manifolds $M, N$ with metrics $g$ and $h$, conformal homeomorphism $M to N$ would be a diffeomorphism $phi : M to N$ such that pullback of $h$ coincides with $g$ multiplied by some function. If $M$ and $N$ are open subsets of complex plane this is equivalent to the condition you quoted (provided that you restrict attention to orientation preserving maps; otherwise you have to include anti-holomorphic functions also).
$endgroup$
– Blazej
Feb 1 at 21:37
|
show 1 more comment
$begingroup$
How much Riemannian geometry do you know?
$endgroup$
– Blazej
Jan 31 at 23:25
$begingroup$
I know what is an homeomorphism and what is a conformal map, but i never faced the conformal homeomorphism
$endgroup$
– Claudio Delfino
Feb 1 at 1:39
$begingroup$
What do would you say a conformal map is?
$endgroup$
– user98602
Feb 1 at 16:32
$begingroup$
For me a conformal map is an $fcolon D to D'$ , where $D,D' subset mathbb{C}$, $f$ analytic and one to one and onto (which should implies that $f'(z)ne0$ for all z and $f^{-1}$ is a conformal map too)
$endgroup$
– Claudio Delfino
Feb 1 at 17:31
$begingroup$
What you refer as a conformal map is a special case. If you have two Riemannian manifolds $M, N$ with metrics $g$ and $h$, conformal homeomorphism $M to N$ would be a diffeomorphism $phi : M to N$ such that pullback of $h$ coincides with $g$ multiplied by some function. If $M$ and $N$ are open subsets of complex plane this is equivalent to the condition you quoted (provided that you restrict attention to orientation preserving maps; otherwise you have to include anti-holomorphic functions also).
$endgroup$
– Blazej
Feb 1 at 21:37
$begingroup$
How much Riemannian geometry do you know?
$endgroup$
– Blazej
Jan 31 at 23:25
$begingroup$
How much Riemannian geometry do you know?
$endgroup$
– Blazej
Jan 31 at 23:25
$begingroup$
I know what is an homeomorphism and what is a conformal map, but i never faced the conformal homeomorphism
$endgroup$
– Claudio Delfino
Feb 1 at 1:39
$begingroup$
I know what is an homeomorphism and what is a conformal map, but i never faced the conformal homeomorphism
$endgroup$
– Claudio Delfino
Feb 1 at 1:39
$begingroup$
What do would you say a conformal map is?
$endgroup$
– user98602
Feb 1 at 16:32
$begingroup$
What do would you say a conformal map is?
$endgroup$
– user98602
Feb 1 at 16:32
$begingroup$
For me a conformal map is an $fcolon D to D'$ , where $D,D' subset mathbb{C}$, $f$ analytic and one to one and onto (which should implies that $f'(z)ne0$ for all z and $f^{-1}$ is a conformal map too)
$endgroup$
– Claudio Delfino
Feb 1 at 17:31
$begingroup$
For me a conformal map is an $fcolon D to D'$ , where $D,D' subset mathbb{C}$, $f$ analytic and one to one and onto (which should implies that $f'(z)ne0$ for all z and $f^{-1}$ is a conformal map too)
$endgroup$
– Claudio Delfino
Feb 1 at 17:31
$begingroup$
What you refer as a conformal map is a special case. If you have two Riemannian manifolds $M, N$ with metrics $g$ and $h$, conformal homeomorphism $M to N$ would be a diffeomorphism $phi : M to N$ such that pullback of $h$ coincides with $g$ multiplied by some function. If $M$ and $N$ are open subsets of complex plane this is equivalent to the condition you quoted (provided that you restrict attention to orientation preserving maps; otherwise you have to include anti-holomorphic functions also).
$endgroup$
– Blazej
Feb 1 at 21:37
$begingroup$
What you refer as a conformal map is a special case. If you have two Riemannian manifolds $M, N$ with metrics $g$ and $h$, conformal homeomorphism $M to N$ would be a diffeomorphism $phi : M to N$ such that pullback of $h$ coincides with $g$ multiplied by some function. If $M$ and $N$ are open subsets of complex plane this is equivalent to the condition you quoted (provided that you restrict attention to orientation preserving maps; otherwise you have to include anti-holomorphic functions also).
$endgroup$
– Blazej
Feb 1 at 21:37
|
show 1 more comment
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$begingroup$
How much Riemannian geometry do you know?
$endgroup$
– Blazej
Jan 31 at 23:25
$begingroup$
I know what is an homeomorphism and what is a conformal map, but i never faced the conformal homeomorphism
$endgroup$
– Claudio Delfino
Feb 1 at 1:39
$begingroup$
What do would you say a conformal map is?
$endgroup$
– user98602
Feb 1 at 16:32
$begingroup$
For me a conformal map is an $fcolon D to D'$ , where $D,D' subset mathbb{C}$, $f$ analytic and one to one and onto (which should implies that $f'(z)ne0$ for all z and $f^{-1}$ is a conformal map too)
$endgroup$
– Claudio Delfino
Feb 1 at 17:31
$begingroup$
What you refer as a conformal map is a special case. If you have two Riemannian manifolds $M, N$ with metrics $g$ and $h$, conformal homeomorphism $M to N$ would be a diffeomorphism $phi : M to N$ such that pullback of $h$ coincides with $g$ multiplied by some function. If $M$ and $N$ are open subsets of complex plane this is equivalent to the condition you quoted (provided that you restrict attention to orientation preserving maps; otherwise you have to include anti-holomorphic functions also).
$endgroup$
– Blazej
Feb 1 at 21:37