# Is there a hyperbolic geometry equivalent to Möbius transformations in spherical geometry?

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}."?

• 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 . – Steven Stadnicki Aug 9 '12 at 0:47
• @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. – Aaron Golden Aug 9 '12 at 19:18
• 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.) – Steven Stadnicki Aug 9 '12 at 21:20