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.
In this talk, I will give a survey on the recent developments on the question about how the local geometric structure affects the global nature of non-Riemannian manifolds with emphasis on the existence problem of compact models of locally symmetric spaces.
T. Kobayashi, On discontinuous group actions on non-Riemannian homogeneous spaces, To appear in Sugaku Expositions, Amer. Math. Soc., math.DG/0603319.
T. Kobayashi and T. Yoshino, Compact Clifford-Klein forms of symmetric spaces — revisited, Pure and Appl. Math. Quarterly 1 (2005), 603-684, Special Issue: In Memory of Armand Borel.
© Toshiyuki Kobayashi