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Geometry and Topology for Lecturers
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Classroom Contents
Hyperbolic Geometry, Fuchsian Groups and Moduli Spaces - Lecture 1
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- 1 Geometry and Topology for Lecturers
- 2 Hyperbolic Geometry, Fuchsian groups and moduli spaces Lecture 1
- 3 Introduction to Hyperbolic Geometry
- 4 1. Upper half-plane model
- 5 Fact 1 Automorphism H2 = PSL2,R
- 6 Fact 2
- 7 Why invariant ?
- 8 Can check
- 9 Properties of the hyperbolic metric
- 10 1. Geodesics
- 11 Consequence
- 12 2. The metric is complete
- 13 3. Sum of interior angles of any geodesic triangle is less than Pi !
- 14 Example of conformal model of the hyperbolic geometry
- 15 In fact
- 16 4. The hyperbolic metric has constant curvature
- 17 2. Disk model
- 18 Note
- 19 Hyperbolic Trigonometry - Warmup
- 20 Lemma
- 21 Proof
- 22 Note: In Euclidean geometry
- 23 3. Hyperboloid model
- 24 Claim
- 25 Example
- 26 Relation with unit disk model
- 27 Q&A
