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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

Description:

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.

Hyperbolic Geometry, Fuchsian Groups and Moduli Spaces - Lecture 1

International Centre for Theoretical Sciences

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#Mathematics #Geometry #Non-Euclidean Geometry #Hyperbolic Geometry #Differential Geometry #Geodesics #Complex Analysis #Riemann Surfaces #Algebraic Geometry #Moduli Space

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