"Title: From Hyperbolic Geometry to Lorentzian Geometry --- Geometry of SL(2,R) ---"
While a Riemannian manifold is equipped with a positive definite metric, a Lorentzian manifold is equipped with an indefinite metric of type (-, +, …,+). Hyperbolic geometry studies manifolds with constant negative curvature. Recently, several connections between hyperbolic geometry and Lorentzian geometry are revealed. This course is intended to introduce these connections. First I introduce 2-dimensional hyperbolic space in the Minkowski space, the flat Lorentzian manifold. Especially I focus on the space of geodesics in the hyperbolic space. The main subject of this course is the anti-de Sitter space, the Lorentzian space with constant negative curvature. This space is Lorentzian analogue of the hyperbolic space. The 3-dimensional anti-de Sitter space can be identified with SL(2,R). The main purpose of this course is to introduce the close connection, revealed by Mess, between the 2-dimensional hyperbolic space and the 3-dimensional anti-de Sitter space. One of the goal is to explain Thurston's Earthquake Theorem in the context of anti-de Sitter geometry.
The aim of this course is to introduce a viewpoint of Lorentzian geometry to study hyperbolic manifolds.
Be familiar with foundation of 2-dimensional hyperbolic geometry.
Be familiar with foundation of 3-dimensional anti-de Sitter geometry.
Be familiar with the notions of Lie groups, homogeneous spaces and Lorentzian geometry.
Hyperbolic geometry, Lorentzian geometry, (pseudo-)Riemannian geometry, the anti-de Sitter space, SL(2,R), homogeneous spaces.
✔ Specialist skills | Intercultural skills | Communication skills | Critical thinking skills | Practical and/or problem-solving skills |
This is a standard lecture course. There will be some assignments.
Course schedule | Required learning | |
---|---|---|
Class 1 | The geometry of the 2-dimensional hyperbolic space: -- the upper half space model, the unit disk model and the hyperboloid model. -- The group of isometries. -- Geodesics on hyperbolic surfaces and earthquake deformations. -- Unit tangent bundles of hyperbolic surfaces and geodesic flows. The geometry of the 3-dimensional anti-de Sitter space: -- Several models, especially, the SL(2,R)-model. -- The group of isometries. -- Geodesics and totally geodesic planes. -- Properly discontinuous action. -- Mess's Theory (an alternate proof for Thurston's Earthquake Theorem). | Details will be provided during each class session |
None required
The one of the aim of this course is to introduce the following paper:
G. Mess, ``Lorentz spacetimes of constant curvature,"" Geom. Dedicata (2007) 3--45.
Assignments (100%).
None Required