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 pseudo-Riemannian geometry, familiar to us as the space-time of relativity theory, and more generally in pseudo-Riemannian geometry of general signature, surprisingly little is known about global properties of the geometry even if we impose a locally homogeneous structure.
In this colloquium, I plan to discuss two topics.
Global geometry: Existence problem of compact manifolds modelled locally on homogeneous spaces, and their deformation theory.
Spectral analysis: Construction of periodic eigenfunctions for the (indefinite) Laplacian, and stability question of eigenvalues under deformation of geometric structure by using representation theory.
© Toshiyuki Kobayashi