Author Archives: aldenwalker

Hyperbolic Geometry Notes #5 – Mostow Rigidity

1. Mostow Rigidity For hyperbolic surfaces, Moduli space is quite large and complicated. However, in three dimensions Moduli space is trivial: Theorem 1 If is a homotopy equivalence of closed hyperbolic manifolds with , then is homotopic to an isometry. … Continue reading

Posted in 3-manifolds, Groups, Hyperbolic geometry, Uncategorized | 2 Comments

Hyperbolic Geometry Notes #4 – Fenchel-Nielsen Coordinates

1. Fenchel-Nielsen Coordinates for Teichmuller Space Here we discuss a very nice set of coordinates for Teichmuller space. The basic idea is that we cut the surface up into small pieces (pairs of pants); hyperbolic structures on these pieces are … Continue reading

Posted in Hyperbolic geometry | 3 Comments

Hyperbolic Geometry Notes #3 – Teichmuller and Moduli Space

This post introduces Teichmuller and Moduli space. The upcoming posts will talk about Fenchel-Nielsen coordinates for Teichmuller space; it’s split up because I figured this was a relatively nice break point. Hopefully, I will later add some pictures to this … Continue reading

Posted in Hyperbolic geometry | 3 Comments

Hyperbolic Geometry Notes #2 – Triangles and Gauss Bonnet

In this post, I will cover triangles and area in spaces of constant (nonzero) curvature. We are focused on hyperbolic space, but we will talk about spheres and the Gauss-Bonnet theorem. 1. Triangles in Hyperbolic Space Suppose we are given … Continue reading

Posted in Euclidean Geometry, Geometric structures, Hyperbolic geometry, Surfaces, Uncategorized | 1 Comment

Hyperbolic Geometry (157b) Notes #1

I am Alden, one of Danny’s students. Error/naivete that may (will) be found here is mine. In these posts, I will attempt to give notes from Danny’s class on hyperbolic geometry (157b). This first post covers some models for hyperbolic … Continue reading

Posted in Commentary, Euclidean Geometry, Groups, Hyperbolic geometry, Lie groups, Overview, Visualization | 5 Comments