Möbius transformations are conformal (i.e., angle-preserving), orientation-preserving, area-preserving isometries in the upper half-plane model of hyperbolic space. It piqued my interest to discover that the form of the metric in this model (which I discussed in… 732 more words
Tags » Hyperbolic Geometry
Hyperbolic Geometry Note #1: Strange behaviour of length calculations in the Poincaré half-plane model of hyperbolic space
In the Poincaré half-plane model of hyperbolic space, the upper half-plane is the set of complex numbers with positive imaginary part:
The boundary of , , is the real axis together with the point . 762 more words
is the unique (up to isometry) complete simply-connected 2-dimensional Riemann manifold of constant sectional curvature -1.
- It is diffeomorphic to as a topological space (or, indeed, isometric as a Riemannian manifold, if we give a left-invariant metric): to show this, we note that has isometry group , and the subgroup of isometries which stabilize any given is isomorphic to . 211 more words