Ma 191j, Real and Complex Hyperbolic Geometry

Spring 2004

Pictures by the Geometry Center and Rich Schwartz respectively.

Course Description

Hyperbolic space, the geometry which has constant negative curvature, is one of the central examples in Riemannian geometry and has deep connections to the theory of 3-manifolds. Complex hyperbolic space is its "complexification", and is a homogeneous geometry of variable negative curvature. In both cases, it is very interesting to try to understand discrete groups of isometries of these geometries, that is, discrete subgroups of the Lie groups SO(1, n) and SU(1, n). Equivalently, we want to understand manifolds with metrics locally isometric to these hyperbolic spaces. When the quotient has finite volume (i.e. when the group is a lattice in the associated Lie group), they are very rigid; Mostow showed that if two such quotients of dim > 2 have the same fundamental group then they are isometric. On the other hand, when the quotient has infinite volume they frequently are quite flexible. Moreover, in this case, it is interesting to understand the dynamics of the action of the group on the sphere at infinity, contrasting the regions where the action is chaotic with those where the action is discrete. I will try to cover the following topics:


Familiarity with smooth manifolds, elementary Riemannian geometry, and a little knowledge of the hyperbolic plane (which you can obtain from reading chapter 2 of Thurston's book listed below).


The course grade will be determined by a final paper. This will be a 6-8 page paper written on a topic in related to the content of this class. The topic will be chosen in consolation with me in the 5th or 6th week of class, and will be due on Friday, June 4. Here are some ideas, with sources:


There will be no class the week of May 3rd. To make up for this, class will continue through June 3rd. Alternatively, extra classes will be scheduled.

Further references

In addition to the sources listed above, there is

W. Goldman, Complex Hyperbolic Geometry Oxford University Press, 1999.

For a good intro to real hyperbolic geometry, see Chapter 2 of

W. Thurston, Three-dimensional geometry and topology, vol 1, Princeton University Press, 1997.

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