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 projects:
Global geometry : Existence problem of compact locally homogeneous spaces, and deformation theory.
Spectral analysis : Construction of periodic eigenfunctions for the Laplacian for indefinitemetric, and discuss the stability of eigenvalued under deformation of geometric structure.
© Toshiyuki Kobayashi