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2017　Special lectures on advanced topics in Mathematics K

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Academic unit or major
Graduate major in Mathematics
Instructor(s)
Kawahira Tomoki  Ito Kentaro 
Class Format
Lecture     
Media-enhanced courses
Day/Period(Room No.)
Intensive ()  
Group
-
Course number
MTH.E535
Credits
2
Academic year
2017
Offered quarter
2Q
Syllabus updated
2017/3/17
Lecture notes updated
-
Language used
Japanese
Access Index

Course description and aims

"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.

Student learning outcomes

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.

Keywords

Hyperbolic geometry, Lorentzian geometry, (pseudo-)Riemannian geometry, the anti-de Sitter space, SL(2,R), homogeneous spaces.

Competencies that will be developed

✔ Specialist skills Intercultural skills Communication skills Critical thinking skills Practical and/or problem-solving skills

Class flow

This is a standard lecture course. There will be some assignments.

Course schedule/Required learning

  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

Textbook(s)

None required

Reference books, course materials, etc.

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.

Assessment criteria and methods

Assignments (100%).

Related courses

  • None

Prerequisites (i.e., required knowledge, skills, courses, etc.)

None Required

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