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?
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.