Recent Developments in Pseudo-Riemannian Geometry
In Riemannian geometry, important progress has been made over the past thirty years in understanding relations between the local and global structure of Riemannian manifolds. Many classification results for different classes of Riemannian manifolds have been obtained: manifolds with additional geometric structure, manifolds satisfying curvature conditions, symmetric and homogeneous Riemannian spaces, etc. Similar results for pseudo-Riemannian manifolds are rare and many problems are still open. Sometimes, one can use a special 'Ansatz' or 'Wick-rotations' to transform problems of pseudo-Riemannian geometry into questions of Riemannian geometry. But in many aspects, pseudo-Riemannian and Riemannian geometry differ essentially. This book is something like a 'proceedings' of the scientific program Geometry of Pseudo-Riemannian Manifolds with Applications in Physics, which was held in Vienna at the Erwin Schrödinger International Institute for Mathematical Physics between September and December of 2005. The book is addressed to advanced students as well as to researchers in differential geometry, global analysis, general relativity and string theory. It shows essential differences between the geometry on manifolds with positive definite metrics and on those with indefinite metrics, and highlights interesting new geometric phenomena arising naturally in the indefinite metric case. The reader can find a description of the present state of the art in the field, as well as open problems, which can stimulate further research.