Hyperbolic distance element

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Most books and websites define the hyperbolic distance element and the corresponding shortest paths in the upper half plane with no explanation. I found a derivation for them in a book called Visual Complex Analysis by Needham and it relied on mapping the "pseudosphere" onto the upper half plane.

Is there an elementary derivation that just uses complex analysis ?
If so, where ?
 

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  • #2
Reb
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Well, if I was to transform the question into the Euclidean setting, I would ask
why would someone need to bother with what a plane is, in order to define straight lines!
What I mean is that there is a minimum of insight required in order to go into the theory of geodesic distance, and this insight is provided by the three model spaces of constant curvature: The sphere, the plane and the pseudosphere.
 
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I hope to know a little differential geometry one day.

There is a book by Ahlfors and Sario that starts off deriving hyperbolic arc length in the unit disk, although I don't understand all the details.
 
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lavinia
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Most books and websites define the hyperbolic distance element and the corresponding shortest paths in the upper half plane with no explanation. I found a derivation for them in a book called Visual Complex Analysis by Needham and it relied on mapping the "pseudosphere" onto the upper half plane.

Is there an elementary derivation that just uses complex analysis ?
If so, where ?

One can develop Lobachevskyan geometry axiomatically. perhaps you should look at this,
 
  • #5
lavinia
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Most books and websites define the hyperbolic distance element and the corresponding shortest paths in the upper half plane with no explanation. I found a derivation for them in a book called Visual Complex Analysis by Needham and it relied on mapping the "pseudosphere" onto the upper half plane.

Is there an elementary derivation that just uses complex analysis ?
If so, where ?

BTW: the importance of the hyperbolic metric is not just that it provides a model for hyperbolic geometry. In the plane of the disc it is invariant under groups of linear fractional transformations whose quotient spaces are all orientable compact surfaces other than the torus and the sphere. This means that any surface of genus greater than 1 has a metric of constant negative curvature. These metrics can not be realized in 3 space and thus are not derivable from surfaces such as the pseudosphere.
 
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I don't know what you mean by not "realizable in 3 space" and by "not derivable from surfaces such as the pseudosphere" and would appreciate a layman's level explanation if possible.

In the Needham book, a conformal mapping is set up between the pseudosphere and the upper half plane. It was nice to see how the hyperbolic elements of arc length and area came from this. That's the sense I might use the word "derivable" in.
 
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  • #7
lavinia
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I don't know what you mean by not "realizable in 3 space" and by "not derivable from surfaces such as the pseudosphere" and would appreciate a layman's level explanation if possible.

A compact surface in 3 space must have a point of positive curvature. These metrics have constant negative curvature and so can not be realized in 3 space.

In the Needham book, a conformal mapping is set up between the pseudosphere and the upper half plane. It was nice to see how the hyperbolic elements of arc length and area came from this. That's the sense I might use the word "derivable" in.

yes but then you have to ask how the pseudosphere was derived.
 
  • #8
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I don't understand the first part but hope to one day.
The second part I understand. It's the reason I want to see a derivation that's self contained in complex analysis. The pseudosphere was intoduced almost out of nowhere in the Visual COmplex Analysis book, as the surface of revolution of a tractrix.
 
  • #9
lavinia
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I don't understand the first part but hope to one day.

Surfaces can have geometries that are not realizable in 3 space. The simplest example is the flat torus which has no curvature but still folds around into a doughnut. However, this surface can be realized 4 space.

Similarly a two holed doughnut can be given a geometry that locally look exactly like the pseudo sphere. with this geometry is can not fit i3 space. i am not what dimension it lives in.
 

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