# Classification of Möbius Transformations

We know how to classify the points on a surface,by looking the Gaussian curvature at a point in order to guess the shape of the surface near that point.On the other hand we classify the Möbius transformations acting on the hyperbolic plane according to their fixed points or equivalently their traces.But we also use analogous names for this classification;elliptic,parabolic and hyperbolic.Is there a connection or is it just conventional?

• More context, more detail, and some citations would help, otherwise the question is rather unclear. For example, are you saying that the term "elliptic point" is used to refer to a point where Gaussian curvature is positive, and if so can you give a citation? – Lee Mosher Jun 8 '16 at 13:25
• I am just wondering that if the terms elliptic ,parabolic and hyperbolic transformation have a basis in the sense of differential geometry or something else. – mathman Jun 8 '16 at 17:16
• Well, their basis is the ancient theorem that classifies conic sections into three classes: ellipses, parabolas, and hyperbolas. Everything since then is analogy. – Lee Mosher Jun 8 '16 at 17:28
• This appears to be related to my own question on The term “elliptic”. – MvG Jun 8 '16 at 21:32

## 1 Answer

There are indeed connexion between the models.

Someone will correct me if I'm wrong but I know of three types of Möbius Transformations that preserve the Unit Disk :

• 1 finite fixed point : elliptic isometry

This corresponds to the euclidian rotation around 1 point when the fixed point is the center of the rotation.

• 2 infinite fixed points : hyperbolic isometry

This corresponds to the euclidian translation. When you apply an euclidian translation there are at least two fixed points. Theses points lie on the lines orthogonal to the translation vector at infinite distance of the space. As there are no such things as vector in hyperbolic spaces, then there are at most two fixed points which lie on geodesic of the hyperbolic isometry at infinite distances.

• 1 infinite fixed point : parabolic isometry

This should corresponds to the degenerated case of 1 finite point and 2 infinite points. But I'm not so sure about this one.

• which type relates to our physical world? – Ooker Nov 13 '17 at 17:57
• @Ooker : For the infinite isometries if our universe is hypebolic then we may find examples at the cosmic scale. For the elliptic isometry that would correspond to going around the same point at constant distance from it, on a hyperbolic surface. – Kii Nov 14 '17 at 9:42