How local geometric structure affects the global nature of manifolds?
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 Lorentz geometry, familiar to us as the space-time of relativity theory, and more generally in pseudo-Riemannian geometry of general signature, surprising little has been known about global properties of the geometry until recently even if we impose a locally homogeneous structure.
I plan to survey this young topic in geometry such as the existence problem of compact locally homogeneous manifolds and their defomation theory.
© Toshiyuki Kobayashi