Non-equivalent metrics on $PSL_2(mathbb{R})$
$begingroup$
I am reading a paper on continued fractions and it uses the following result on Lie Groups:
Fix an arbitrary left-invariant metric $d$ on $PSL_2(mathbb{R})$ ...
This phrase really throws me off... How many such left-invariant metrics are there? Consider two maps,
$$ z mapsto frac{a_0z+b_0}{c_0z+d_0} , hspace{0.25in}z mapsto frac{a_1z+b_1}{c_1z+d_1} $$
each with $ad-bc=1$, what is the appropriate notion of distance here?
Can someone illustrate two non-equivalent left-invariant metrics on the space of Fractional Linear Transformations?
If it is easier, I will mysteriously ask for different (non-equivalent) metrics on $T^1(mathbb{H}) simeq PSL_2(mathbb{R})$
These are not points in the Hyperbolic plane, these are geodesics in the hyperbolic plane. In the picture below you can see semi-circles in the checkered pattern below, these can be identified with unit tangent vectors
related
- Expression of the Hyperbolic Distance in the Upper Half Plane
- Description of SU(1, 1)
- What are elements in $SU(1, 1)$?
There's some question of what I mean by "equivalent" "metric" (along the lines of Bill Clinton's what does "is" mean?) I am just trying to see if there are really any "different" left-invariant metrics.
Example of equivalent but not strongly equivalent metrics
Understanding equivalent metric spaces
Left invariant is pretty restrictive. I believe in the case of $mathbb{H}$ hyperbolic plane there is only one translation invariant constant curvature metric up to scalar multiplication (so we can set curvature to -1).
My bst guess for $T^1(mathbb{H})$ is $frac{dx^2 + dy^2}{y^2} + dtheta^2 $ where $theta$ is the angle of the unit tangent vector.
Is the left-invariant metric on $T^1(mathbb{H})=PSL_2(mathbb{R})$ unique up to scalar multiplication ?
lie-groups hyperbolic-geometry
asked Jan 14 '16 at 16:48
cactus314cactus314
15.4k42269
$endgroup$
add a comment |
$begingroup$
I am reading a paper on continued fractions and it uses the following result on Lie Groups:
Fix an arbitrary left-invariant metric $d$ on $PSL_2(mathbb{R})$ ...
This phrase really throws me off... How many such left-invariant metrics are there? Consider two maps,
$$ z mapsto frac{a_0z+b_0}{c_0z+d_0} , hspace{0.25in}z mapsto frac{a_1z+b_1}{c_1z+d_1} $$
each with $ad-bc=1$, what is the appropriate notion of distance here?
Can someone illustrate two non-equivalent left-invariant metrics on the space of Fractional Linear Transformations?
If it is easier, I will mysteriously ask for different (non-equivalent) metrics on $T^1(mathbb{H}) simeq PSL_2(mathbb{R})$
These are not points in the Hyperbolic plane, these are geodesics in the hyperbolic plane. In the picture below you can see semi-circles in the checkered pattern below, these can be identified with unit tangent vectors
related
- Expression of the Hyperbolic Distance in the Upper Half Plane
- Description of SU(1, 1)
- What are elements in $SU(1, 1)$?
There's some question of what I mean by "equivalent" "metric" (along the lines of Bill Clinton's what does "is" mean?) I am just trying to see if there are really any "different" left-invariant metrics.
Example of equivalent but not strongly equivalent metrics
Understanding equivalent metric spaces
Left invariant is pretty restrictive. I believe in the case of $mathbb{H}$ hyperbolic plane there is only one translation invariant constant curvature metric up to scalar multiplication (so we can set curvature to -1).
My bst guess for $T^1(mathbb{H})$ is $frac{dx^2 + dy^2}{y^2} + dtheta^2 $ where $theta$ is the angle of the unit tangent vector.
Is the left-invariant metric on $T^1(mathbb{H})=PSL_2(mathbb{R})$ unique up to scalar multiplication ?
lie-groups hyperbolic-geometry
asked Jan 14 '16 at 16:48
cactus314cactus314
15.4k42269
$endgroup$
$begingroup$
Two questions: what do you mean by "metric," and what do you mean by "equivalent" here?
$endgroup$
– Daniel McLaury
Jan 14 '16 at 23:30
add a comment |
$begingroup$
I am reading a paper on continued fractions and it uses the following result on Lie Groups:
Fix an arbitrary left-invariant metric $d$ on $PSL_2(mathbb{R})$ ...
This phrase really throws me off... How many such left-invariant metrics are there? Consider two maps,
$$ z mapsto frac{a_0z+b_0}{c_0z+d_0} , hspace{0.25in}z mapsto frac{a_1z+b_1}{c_1z+d_1} $$
each with $ad-bc=1$, what is the appropriate notion of distance here?
Can someone illustrate two non-equivalent left-invariant metrics on the space of Fractional Linear Transformations?
If it is easier, I will mysteriously ask for different (non-equivalent) metrics on $T^1(mathbb{H}) simeq PSL_2(mathbb{R})$
These are not points in the Hyperbolic plane, these are geodesics in the hyperbolic plane. In the picture below you can see semi-circles in the checkered pattern below, these can be identified with unit tangent vectors
related
- Expression of the Hyperbolic Distance in the Upper Half Plane
- Description of SU(1, 1)
- What are elements in $SU(1, 1)$?
There's some question of what I mean by "equivalent" "metric" (along the lines of Bill Clinton's what does "is" mean?) I am just trying to see if there are really any "different" left-invariant metrics.
Example of equivalent but not strongly equivalent metrics
Understanding equivalent metric spaces
Left invariant is pretty restrictive. I believe in the case of $mathbb{H}$ hyperbolic plane there is only one translation invariant constant curvature metric up to scalar multiplication (so we can set curvature to -1).
My bst guess for $T^1(mathbb{H})$ is $frac{dx^2 + dy^2}{y^2} + dtheta^2 $ where $theta$ is the angle of the unit tangent vector.
Is the left-invariant metric on $T^1(mathbb{H})=PSL_2(mathbb{R})$ unique up to scalar multiplication ?
lie-groups hyperbolic-geometry
asked Jan 14 '16 at 16:48
cactus314cactus314
15.4k42269
$endgroup$
I am reading a paper on continued fractions and it uses the following result on Lie Groups:
Fix an arbitrary left-invariant metric $d$ on $PSL_2(mathbb{R})$ ...
This phrase really throws me off... How many such left-invariant metrics are there? Consider two maps,
$$ z mapsto frac{a_0z+b_0}{c_0z+d_0} , hspace{0.25in}z mapsto frac{a_1z+b_1}{c_1z+d_1} $$
each with $ad-bc=1$, what is the appropriate notion of distance here?
Can someone illustrate two non-equivalent left-invariant metrics on the space of Fractional Linear Transformations?
If it is easier, I will mysteriously ask for different (non-equivalent) metrics on $T^1(mathbb{H}) simeq PSL_2(mathbb{R})$
These are not points in the Hyperbolic plane, these are geodesics in the hyperbolic plane. In the picture below you can see semi-circles in the checkered pattern below, these can be identified with unit tangent vectors
related
- Expression of the Hyperbolic Distance in the Upper Half Plane
- Description of SU(1, 1)
- What are elements in $SU(1, 1)$?
There's some question of what I mean by "equivalent" "metric" (along the lines of Bill Clinton's what does "is" mean?) I am just trying to see if there are really any "different" left-invariant metrics.
Example of equivalent but not strongly equivalent metrics
Understanding equivalent metric spaces
Left invariant is pretty restrictive. I believe in the case of $mathbb{H}$ hyperbolic plane there is only one translation invariant constant curvature metric up to scalar multiplication (so we can set curvature to -1).
My bst guess for $T^1(mathbb{H})$ is $frac{dx^2 + dy^2}{y^2} + dtheta^2 $ where $theta$ is the angle of the unit tangent vector.
Is the left-invariant metric on $T^1(mathbb{H})=PSL_2(mathbb{R})$ unique up to scalar multiplication ?
lie-groups hyperbolic-geometry
lie-groups hyperbolic-geometry
asked Jan 14 '16 at 16:48
cactus314cactus314
15.4k42269
asked Jan 14 '16 at 16:48
cactus314cactus314
15.4k42269
edited Dec 12 '18 at 21:18
cactus314
asked Jan 14 '16 at 16:48
cactus314cactus314
15.4k42269
asked Jan 14 '16 at 16:48
cactus314cactus314
15.4k42269
asked Jan 14 '16 at 16:48
cactus314cactus314
15.4k42269
15.4k42269
$begingroup$
Two questions: what do you mean by "metric," and what do you mean by "equivalent" here?
$endgroup$
– Daniel McLaury
Jan 14 '16 at 23:30
add a comment |
$begingroup$
Two questions: what do you mean by "metric," and what do you mean by "equivalent" here?
$endgroup$
– Daniel McLaury
Jan 14 '16 at 23:30
$begingroup$
Two questions: what do you mean by "metric," and what do you mean by "equivalent" here?
$endgroup$
– Daniel McLaury
Jan 14 '16 at 23:30
$begingroup$
Two questions: what do you mean by "metric," and what do you mean by "equivalent" here?
$endgroup$
– Daniel McLaury
Jan 14 '16 at 23:30
add a comment |
1 Answer
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You can multiply any left-invariant metric by a positive number and you will get a different left-invariant metric.
answered Jan 15 '16 at 15:34
Lee MosherLee Mosher
49.1k33685
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$begingroup$
You can multiply any left-invariant metric by a positive number and you will get a different left-invariant metric.
answered Jan 15 '16 at 15:34
Lee MosherLee Mosher
49.1k33685
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add a comment |
$begingroup$
You can multiply any left-invariant metric by a positive number and you will get a different left-invariant metric.
answered Jan 15 '16 at 15:34
Lee MosherLee Mosher
49.1k33685
$endgroup$
add a comment |
$begingroup$
You can multiply any left-invariant metric by a positive number and you will get a different left-invariant metric.
answered Jan 15 '16 at 15:34
Lee MosherLee Mosher
49.1k33685
$endgroup$
You can multiply any left-invariant metric by a positive number and you will get a different left-invariant metric.
answered Jan 15 '16 at 15:34
Lee MosherLee Mosher
49.1k33685
answered Jan 15 '16 at 15:34
Lee MosherLee Mosher
49.1k33685
answered Jan 15 '16 at 15:34
Lee MosherLee Mosher
49.1k33685
answered Jan 15 '16 at 15:34
Lee MosherLee Mosher
49.1k33685
49.1k33685
add a comment |
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Two questions: what do you mean by "metric," and what do you mean by "equivalent" here?
$endgroup$
– Daniel McLaury
Jan 14 '16 at 23:30