Hyperbolic Geometry, Fuchsian Groups and Moduli Spaces - Lecture 1

Explore the foundations of hyperbolic geometry in this comprehensive lecture from the "Geometry and Topology for Lecturers" workshop. Delve into the upper half-plane model, examining its automorphisms and properties of the hyperbolic metric. Investigate geodesics, completeness, and the unique characteristics of hyperbolic triangles. Compare the disk model and hyperboloid model, uncovering their relationships and distinctions. Engage with hyperbolic trigonometry and gain insights into the constant curvature of hyperbolic space. Perfect for mathematics lecturers and researchers seeking to deepen their understanding of non-Euclidean geometries and their applications in topology and moduli spaces.

Syllabus

Geometry and Topology for Lecturers
Hyperbolic Geometry, Fuchsian groups and moduli spaces Lecture 1
Introduction to Hyperbolic Geometry
1. Upper half-plane model
Fact 1 Automorphism H2 = PSL2,R
Fact 2
Why invariant ?
Can check
Properties of the hyperbolic metric
1. Geodesics
Consequence
2. The metric is complete
3. Sum of interior angles of any geodesic triangle is less than Pi !
Example of conformal model of the hyperbolic geometry
In fact
4. The hyperbolic metric has constant curvature
2. Disk model
Note
Hyperbolic Trigonometry - Warmup
Lemma
Proof
Note: In Euclidean geometry
3. Hyperboloid model
Claim
Example
Relation with unit disk model
Q&A