Global Lorentzian Geometry

Winter 2007


Syllabus

Mathematics 250263


Instructor

Dr. James D.E. Grant
james.grant@univie.ac.at
Fakultät für Mathematik
Universität Wien
Wien

Telephone01 4277 50632
Fax01 4277 50620

Aim of the Course

To introduce some of the main ideas and tools in the modern study of global properties (e.g. completeness, compactness etc) of Lorentzian and Riemannian manifolds. Particular goals are to discuss, in some depth, the following topics:

  1. Singularity theorems in Lorentzian geometry;
  2. Comparison theorems (e.g. the Bishop-Gromov comparison theorem) in Riemannian geometry;
  3. The positive mass theorem in Lorentzian and Riemannian geometry (time permitting).

Prerequisites

It is assumed that you will be familiar with the material contained in the courses Differential Geometry I and Differential Geometry II.

Lectures

Day Time Room
Wednesday 1-3pm D 101
Thursday 1-3pm D 101

Please note that this course will be taught in English.


Office hours

I will be in my office, C 237, Monday 9-10am. You may drop by then to ask questions. If you wish to see me at any other time, please email me so that we can make an appointment.

Textbooks

The course should be quite self-contained, so there should no need for you to buy a textbook. I will also type up a set of lecture notes, which you will be able to download from these web-pages.

We will use material from the following books on Lorentzian and semi-Riemannian geometry, which you may like to glance through:

  1. S.W. Hawking and G. R. Ellis, The Large Scale Structure of Space-Time, Cambridge University Press, Cambridge, 1975.

  2. B. O'Neill, Semi-Riemannian geometry, Pure and Applied Mathematics, vol. 103, Academic Press Inc.
  3. J.K. Beem, P.E. Ehrlich, and K.L. Easley, Global Lorentzian geometry, Monographs and Textbooks in Pure and Applied Mathematics, vol. 202, Marcel Dekker Inc., New York, 1996. (Out of print, as far as I know.)

We will also use some material from the following books on Riemannian and differential geometry:

  1. I. Chavel, Riemannian geometry, Cambridge Studies in Advanced Mathematics, vol.~98, Cambridge University Press, Cambridge, 2006.
  2. S. Gallot, D. Hulin and J. Lafontaine, Riemannian geometry, Universitext, Springer-Verlag, Berlin, 2004.
  3. F.W. Warner, Foundations of differentiable manifolds and Lie groups, Graduate Texts in Mathematics, vol.~94, Springer-Verlag, New York, 1983.

The following resources offer background material (and more) on differential geometry. They are also free.

  1. P.W. Michor, Topics in Differential Geometry, 2006.
  2. A. Kriegl, Differentialgeometrie, 2007.