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, as well as in various other kinds of geometry (symplectic, complex geometry, ...), surprising little is known about global properties of the geometry even if we impose a locally homogeneous structure.I will give a survey on the recent developments regarding the question about how the local geometric structure affects the global nature of non-Riemannian manifolds, with emphasis on the existence problem of compact forms, rigidity and deformation.
© Toshiyuki Kobayashi