Hyperbolic Geometry, Fuchsian Groups and Moduli Spaces - Lecture 1
International Centre for Theoretical Sciences via YouTube
Overview
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
Taught by
International Centre for Theoretical Sciences