Special Research Semester at the International Erwin Schrödinger Institute for Mathematical Physics (ESI) Vienna, Austria

Geometry of pseudo-Riemannian manifolds with application in Physics

      September – December 2005

Aim: The aim of the Special Semester is to bring together highly qualified researchers working on different aspects of the geometry of manifolds with indefinite metrics. Whereas in Riemannian geometry  in the last 30 years an essential progress was made in the investigation and classification of different classes of Riemannian manifolds (e.g. manifolds with an addi-tional geometric structures, manifolds with conditions on curvature, homogeneous Riemannian spaces), similar results for pseudo-Riemannian manifolds are rare and many problems are still open. For a long time the main source of problems in pseudo-Riemannian geometry was General Relativity which deals with 4-dimensional Lorentzian manifolds (space-times). However, the developments in Theoretical Physics (Supergravity, String theory) require a deeper understanding of the geometric structure of higher dimensional manifolds with indefinite metrics of different signature. Sometimes, one can use special Ansatzes or "Wick-rotations" to transform the problems of pseudo-Riemannian geometry  into  Riemannian ones. But in many aspects pseudo-Riemannian geometry essentially differs from Riemannian one  and many specific, highly nontrivial and interesting questions appear. In the last time different groups started to focus their attention to geometric problems arising by indefinite metrics. During the program we want to discuss the results of different groups, we want to develop the field by joint research during the Special Semester and we want to establish a closer cooperation between the international groups.

• Classification of  holonomy representations of pseudo-Riemannian manifolds
• Global models of pseudo-Riemannian manifolds with special holonomy
• Symmetries of pseudo-Riemannian manifolds, symmetric and homogeneous spaces, pseudo-Riemannian G-manifolds and their orbit structures.
• Special pseudo-Riemannian geometries and their relation to other geometric structures
• Spin geometry of pseudo-Riemannian manifolds
• Einstein equation and integrable systems in pseudo-Riemannian geometry
• Conformal geometry in indefinite signature (Kleinian spaces, conformal invariants, conformal completions)
• Geometry of submanifolds in spaces with indefinite metrics
• Dynamics of geodesics in pseudo-Riemannian spaces
• Applications in Supergravity  and String theory

We plan to publish Proceedings of the Special Semester which on the one hand shall give a pedagogical introduction into the subject and which on the other hand shall describe the state of art of the field.
 

 

• d.v.alekseevsky@hull.ac.uk
• baum@mathematik.hu-berlin.de