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 periodic eigenfunctions for the Laplacian for indefinite-metric, and discuss the stability of eigenvalued under the deformation of the geometric structure.
by taking anti-de Sitter manifolds as a typical example.
© Toshiyuki Kobayashi