Geometry: Hyperbolic space - the upper half-space and Poincaré disk models. 2nd Year Student Lecture

Overview

This lecture introduces two models of hyperbolic space: the upper half-space and the Poincaré disk, demonstrating their equivalence to the hyperboloid model via isometries. It details the metric, isometry group (Mobius group), and geodesics for both the upper half-space (vertical lines and semicircles) and the Poincaré disk (lines and circles through the origin/boundary). The lecture highlights surprising properties like dilation invariance and the non-intuitive behavior of distances near boundaries.

Key takeaways

The hyperboloid model, upper half-space model, and Poincaré disk model are all isometric representations of hyperbolic space.
The distance metric in hyperbolic space causes distances to grow infinitely large as one approaches the boundary of the space.
The isometry group for both the upper half-space and the Poincaré disk is the Mobius group, comprising specific transformations preserving the metric.
Geodesics in the upper half-space are vertical lines and semicircles meeting the real axis orthogonally.
Geodesics in the Poincaré disk are arcs of circles meeting the boundary circle orthogonally, including straight lines passing through the origin.
Dilation (scaling) is an isometry in hyperbolic space, meaning distances remain invariant under uniform scaling of the entire space.

Chapters

0:12 Mapping the Hyperboloid to the Upper Half-Space

Defines a map T from the hyperboloid H2 in R^1,2 to the complex plane C.
The map T(X0, X1, X2) = (-X2 + i) / (X0 - X1) projects points onto the upper half-space (Im(z) > 0).
This map is a bijection, allowing the transfer of the metric from the hyperboloid to the upper half-space.

8:21 Defining the Metric on the Upper Half-Space

The distance in the upper half-space is derived from the hyperbolic metric on H2 via the inverse map.
The distance D(z1, z2) = arccosh(1 + |z1 - z2|^2 / (2 * Im(z1) * Im(z2))) is established.
This metric space is shown to be an isometry with the hyperboloid model.

12:38 Distance Example: Points on the Imaginary Axis

Calculates the distance between two points i*y1 and i*y2 on the imaginary axis.
The distance simplifies to log(y2/y1) for y2 > y1.
Demonstrates that distances tend to infinity as points approach the real axis (y -> 0), implying the boundary is never reached.

22:11 Isometry Group of the Upper Half-Space

The orientation-preserving isometry group of the upper half-space H2 is the Mobius group.
This group consists of transformations z -> (az+b)/(cz+d) where a,b,c,d are real and ad-bc=1 (PSL(2,R)).
The full isometry group includes orientation-reversing maps like z -> -z_bar.

29:13 Examples of Isometries in the Upper Half-Space

Translations (z -> z+b) are orientation-preserving isometries.
Dilations (z -> a*z for real a != 0) are also orientation-preserving isometries, a surprising result compared to Euclidean space.
These isometries preserve the hyperbolic metric.

32:34 Geodesics of the Upper Half-Space

Geodesics are images of hyperboloid geodesics under the isometry map T.
They consist of vertical half-lines (Re(z) = constant) and semicircles centered on the real axis.
These curves meet the real axis orthogonally.

42:34 Mapping the Hyperboloid to the Poincaré Disk

Introduces the Poincaré disk model D as another representation of hyperbolic space.
A map T(X0, X1, X2) = (X1 + i*X2) / (1 + X0) projects points from the hyperboloid into the unit disk |z| < 1.
This map is a bijection, enabling metric transfer.

Summary, takeaways, and chapters were generated by AI from the video's transcript and may contain errors. The video belongs to its creator, Oxford Mathematics.