There is a sense in which all "interesting" properties of functions in spherical geometry are invariant under conjugation by a Möbius transformation. The reason is that the Möbius transformations correspond to "uninteresting" manipulations of the whole sphere, as illustrated in this video.

Is there an equivalent notion in hyperbolic geometry? In other words, is there a valid statement of the form "All interesting properties of functions in hyperbolic geometry are invariant under conjugation by a $\text{Foo}$ because the $\text{Foo}$s correspond to uninteresting manipulations of the {upper half plane, unit disk, hyperboloid, etc}."?

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    $\begingroup$ It sounds like what you're looking for is the notion of a (conformal) isometry - see, for instance, geom.uiuc.edu/~crobles/hyperbolic/hypr/isom . $\endgroup$ – Steven Stadnicki Aug 9 '12 at 0:47
  • $\begingroup$ @StevenStadnicki: I think isometries might be a little too restrictive because conjugation by a dilation should preserve interesting properties of a function. In fact, I suspect that dilations correspond to raising or lowering the hyperboloid in the hyperboloid model. $\endgroup$ – Aaron Golden Aug 9 '12 at 19:18
  • $\begingroup$ The problem is that dilations in the hyperbolic plane aren't conformal (for instance, consider the relationship between area and angle defect in hyperbolic triangles); while it's not clear what 'interesting properties' you're interested in preserving, conformality would seem to be an essential one. (This lack of conformality under dilation, incidentally, is why Penrose-style aperiodic self-similar tilings don't work on the hyperbolic plane.) $\endgroup$ – Steven Stadnicki Aug 9 '12 at 21:20

There are two cases to consider. The distinction might be best expressed in terms of that video. There are those transformations which rotate the sphere but leave it in its place. These are certainly uninteresting in the sense that the geometry on the sphere is the same, only its orientation has changed. But then there are also those transformations which move the sphere in space. When you take a unit sphere lying on the plane, then move that sphere in space, then project onto the plane and from there project back to the original unit sphere (instead of the translated one), then the result will be quite different. For example, great circles will map to circles which are not great circles. I'm not sure I'd call this an uninteresting transformation.

The first class, which is really uninteresting, consists of isometries on the sphere. Its equivalent are isometries of the hyperbolic space, which in the Poincaré disk model are those Möbius transformations which fix the unit circle. On the other hand, the second class consists of conformal maps which are not isometries. In the spherical case, the set of all conformal maps of the (Riemann) sphere corresponds exactly to the set of all Möbius transformations. In the hyperbolic case, you again look at the Poincaré disk model, which is conformal in the sense that hyperbolic and Euclidean angle agree. The set of points is the interior of the unit disk, and this set has to be mapped onto itself. The set of conformal transformations of the unit disk are exactly the Möbius transformations which fix the unit disk, and therefore also fix the unit circle. This is a consequence of the Riemann mapping theorem, which states that any topological disk can be mapped conformally to the unit disk, and that this map is unique up to Möbius transformation. So the unit disk maps onto itself in a way which is unique up to a Möbius transformation preserving the circle, and since one possible map is the identity, the set of all maps are the Möbius transformations preserving the circle.

So to state this concisely: There are no conformal maps of the hyperbolic plane onto itself which are not hyperbolic isometries. I guess someone else might have made this statement before, but I have neither a name nor a reference for this at the moment.

Note that the whole discussion above should probably include anti Möbius transformations along with Möbius transformations in many places, in order to include operations which reverse all angles into account as well. Particularly since these are usually included with the isometries. But this changes nothing about the core of this discussion.


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