Mixing
Hyperbolic plane via parametrisation of the pseudosphere
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My lecture notes (http://www.matematik.lu.se/matematiklu/personal/sigma/Gauss.pdf),
introduces the hyperbolic plane via a fundamental form on the pseudosphere that induces a metric on the parameterspace.
As far as I understand or can tell, this metric depends pointwise on which point your are in the plane, making things awkward to me. A metric should measure distances between two points and not a scalarproduct at each point in the plane.
The ideas starts with "Remark $4.37$" at page $36$ and then the plane is introduced on page $66$.
Does anyone see where I go wrong?
differential-geometry hyperbolic-geometry
asked Jan 20 at 17:09
MaxedMaxed
2421527
$endgroup$
|
show 2 more comments
$begingroup$
My lecture notes (http://www.matematik.lu.se/matematiklu/personal/sigma/Gauss.pdf),
introduces the hyperbolic plane via a fundamental form on the pseudosphere that induces a metric on the parameterspace.
As far as I understand or can tell, this metric depends pointwise on which point your are in the plane, making things awkward to me. A metric should measure distances between two points and not a scalarproduct at each point in the plane.
The ideas starts with "Remark $4.37$" at page $36$ and then the plane is introduced on page $66$.
Does anyone see where I go wrong?
differential-geometry hyperbolic-geometry
asked Jan 20 at 17:09
MaxedMaxed
2421527
$endgroup$
$begingroup$
Are you familiar with the concept of Riemannian metric?
$endgroup$
– Amitai Yuval
Jan 20 at 18:54
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@AmitaiYuval no, not more then is written here mathworld.wolfram.com/RiemannianMetric.html
$endgroup$
– Maxed
Jan 20 at 18:57
3
$begingroup$
So, basically, what you describe is how things work with a Riemannian metric. You define an inner product at each point, which yields the norm of a tangent vector. Then, by integration, you define the length of a curve. Finally, you define the induced distance between two points as the infimum of lengths of all curves connecting these points. This is a fundamental notion in differential geometry, you should grab a textbook and read about it. Lee's Introduction to Smooth Manifolds is a fair option, as well as many other textbooks.
$endgroup$
– Amitai Yuval
Jan 20 at 19:36
1
$begingroup$
Just to make the point explicit, although you express the thought that "a metric should [not measure] a scalar product at each point in the plane", Riemann's great advance was to think of a metric exactly like that.
$endgroup$
– Lee Mosher
Jan 20 at 21:56
1
$begingroup$
Yes, getting the actual metric from the inner product involves an optimization, namely the infimum described in the comment of @AmitaiYuval.
$endgroup$
– Lee Mosher
Jan 21 at 15:03
|
show 2 more comments
$begingroup$
My lecture notes (http://www.matematik.lu.se/matematiklu/personal/sigma/Gauss.pdf),
introduces the hyperbolic plane via a fundamental form on the pseudosphere that induces a metric on the parameterspace.
As far as I understand or can tell, this metric depends pointwise on which point your are in the plane, making things awkward to me. A metric should measure distances between two points and not a scalarproduct at each point in the plane.
The ideas starts with "Remark $4.37$" at page $36$ and then the plane is introduced on page $66$.
Does anyone see where I go wrong?
differential-geometry hyperbolic-geometry
asked Jan 20 at 17:09
MaxedMaxed
2421527
$endgroup$
My lecture notes (http://www.matematik.lu.se/matematiklu/personal/sigma/Gauss.pdf),
introduces the hyperbolic plane via a fundamental form on the pseudosphere that induces a metric on the parameterspace.
As far as I understand or can tell, this metric depends pointwise on which point your are in the plane, making things awkward to me. A metric should measure distances between two points and not a scalarproduct at each point in the plane.
The ideas starts with "Remark $4.37$" at page $36$ and then the plane is introduced on page $66$.
Does anyone see where I go wrong?
differential-geometry hyperbolic-geometry
differential-geometry hyperbolic-geometry
asked Jan 20 at 17:09
MaxedMaxed
2421527
asked Jan 20 at 17:09
MaxedMaxed
2421527
asked Jan 20 at 17:09
MaxedMaxed
2421527
asked Jan 20 at 17:09
MaxedMaxed
2421527
asked Jan 20 at 17:09
MaxedMaxed
2421527
2421527
$begingroup$
Are you familiar with the concept of Riemannian metric?
$endgroup$
– Amitai Yuval
Jan 20 at 18:54
$begingroup$
@AmitaiYuval no, not more then is written here mathworld.wolfram.com/RiemannianMetric.html
$endgroup$
– Maxed
Jan 20 at 18:57
3
$begingroup$
So, basically, what you describe is how things work with a Riemannian metric. You define an inner product at each point, which yields the norm of a tangent vector. Then, by integration, you define the length of a curve. Finally, you define the induced distance between two points as the infimum of lengths of all curves connecting these points. This is a fundamental notion in differential geometry, you should grab a textbook and read about it. Lee's Introduction to Smooth Manifolds is a fair option, as well as many other textbooks.
$endgroup$
– Amitai Yuval
Jan 20 at 19:36
1
$begingroup$
Just to make the point explicit, although you express the thought that "a metric should [not measure] a scalar product at each point in the plane", Riemann's great advance was to think of a metric exactly like that.
$endgroup$
– Lee Mosher
Jan 20 at 21:56
1
$begingroup$
Yes, getting the actual metric from the inner product involves an optimization, namely the infimum described in the comment of @AmitaiYuval.
$endgroup$
– Lee Mosher
Jan 21 at 15:03
|
show 2 more comments
$begingroup$
Are you familiar with the concept of Riemannian metric?
$endgroup$
– Amitai Yuval
Jan 20 at 18:54
$begingroup$
@AmitaiYuval no, not more then is written here mathworld.wolfram.com/RiemannianMetric.html
$endgroup$
– Maxed
Jan 20 at 18:57
3
$begingroup$
So, basically, what you describe is how things work with a Riemannian metric. You define an inner product at each point, which yields the norm of a tangent vector. Then, by integration, you define the length of a curve. Finally, you define the induced distance between two points as the infimum of lengths of all curves connecting these points. This is a fundamental notion in differential geometry, you should grab a textbook and read about it. Lee's Introduction to Smooth Manifolds is a fair option, as well as many other textbooks.
$endgroup$
– Amitai Yuval
Jan 20 at 19:36
1
$begingroup$
Just to make the point explicit, although you express the thought that "a metric should [not measure] a scalar product at each point in the plane", Riemann's great advance was to think of a metric exactly like that.
$endgroup$
– Lee Mosher
Jan 20 at 21:56
1
$begingroup$
Yes, getting the actual metric from the inner product involves an optimization, namely the infimum described in the comment of @AmitaiYuval.
$endgroup$
– Lee Mosher
Jan 21 at 15:03
$begingroup$
Are you familiar with the concept of Riemannian metric?
$endgroup$
– Amitai Yuval
Jan 20 at 18:54
$begingroup$
Are you familiar with the concept of Riemannian metric?
$endgroup$
– Amitai Yuval
Jan 20 at 18:54
$begingroup$
@AmitaiYuval no, not more then is written here mathworld.wolfram.com/RiemannianMetric.html
$endgroup$
– Maxed
Jan 20 at 18:57
$begingroup$
@AmitaiYuval no, not more then is written here mathworld.wolfram.com/RiemannianMetric.html
$endgroup$
– Maxed
Jan 20 at 18:57
3
3
$begingroup$
So, basically, what you describe is how things work with a Riemannian metric. You define an inner product at each point, which yields the norm of a tangent vector. Then, by integration, you define the length of a curve. Finally, you define the induced distance between two points as the infimum of lengths of all curves connecting these points. This is a fundamental notion in differential geometry, you should grab a textbook and read about it. Lee's Introduction to Smooth Manifolds is a fair option, as well as many other textbooks.
$endgroup$
– Amitai Yuval
Jan 20 at 19:36
$begingroup$
So, basically, what you describe is how things work with a Riemannian metric. You define an inner product at each point, which yields the norm of a tangent vector. Then, by integration, you define the length of a curve. Finally, you define the induced distance between two points as the infimum of lengths of all curves connecting these points. This is a fundamental notion in differential geometry, you should grab a textbook and read about it. Lee's Introduction to Smooth Manifolds is a fair option, as well as many other textbooks.
$endgroup$
– Amitai Yuval
Jan 20 at 19:36
1
1
$begingroup$
Just to make the point explicit, although you express the thought that "a metric should [not measure] a scalar product at each point in the plane", Riemann's great advance was to think of a metric exactly like that.
$endgroup$
– Lee Mosher
Jan 20 at 21:56
$begingroup$
Just to make the point explicit, although you express the thought that "a metric should [not measure] a scalar product at each point in the plane", Riemann's great advance was to think of a metric exactly like that.
$endgroup$
– Lee Mosher
Jan 20 at 21:56
1
1
$begingroup$
Yes, getting the actual metric from the inner product involves an optimization, namely the infimum described in the comment of @AmitaiYuval.
$endgroup$
– Lee Mosher
Jan 21 at 15:03
$begingroup$
Yes, getting the actual metric from the inner product involves an optimization, namely the infimum described in the comment of @AmitaiYuval.
$endgroup$
– Lee Mosher
Jan 21 at 15:03
|
show 2 more comments
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$begingroup$
Are you familiar with the concept of Riemannian metric?
$endgroup$
– Amitai Yuval
Jan 20 at 18:54
$begingroup$
@AmitaiYuval no, not more then is written here mathworld.wolfram.com/RiemannianMetric.html
$endgroup$
– Maxed
Jan 20 at 18:57
3
$begingroup$
So, basically, what you describe is how things work with a Riemannian metric. You define an inner product at each point, which yields the norm of a tangent vector. Then, by integration, you define the length of a curve. Finally, you define the induced distance between two points as the infimum of lengths of all curves connecting these points. This is a fundamental notion in differential geometry, you should grab a textbook and read about it. Lee's Introduction to Smooth Manifolds is a fair option, as well as many other textbooks.
$endgroup$
– Amitai Yuval
Jan 20 at 19:36
1
$begingroup$
Just to make the point explicit, although you express the thought that "a metric should [not measure] a scalar product at each point in the plane", Riemann's great advance was to think of a metric exactly like that.
$endgroup$
– Lee Mosher
Jan 20 at 21:56
1
$begingroup$
Yes, getting the actual metric from the inner product involves an optimization, namely the infimum described in the comment of @AmitaiYuval.
$endgroup$
– Lee Mosher
Jan 21 at 15:03