The local to global study of geometries was a major trend of 20th century geometry, with remarkable developments achieved particularly in Riemannian geometry. In contrast, in areas such as preudo-Riemannian geometry, familiar to us as the spacetime of relativity theory, and more generally in pseudo-Riemannian geometry of general signature, surprising little is known about global properties of the geometry even if we impose a locally homogeneous structure.I plan to explain two programs:

1. (global shape) Existence problem of compact locally homogeneous spaces, and deformation theory.

2. (spectral analysis) Construction of the spectrum of the Laplacian, and discuss its stability under the deformation of the geometric structure.

by taking anti-de Sitter manifolds as a typical example.

© Toshiyuki Kobayashi