## *Global Geometry on Locally Symmetric Spaces—beyond the
Riemannian case*.
Representations of Lie
groups and applications (Scientific committee: L. Clozel, N. Lohoué, R. Parthasarathy and M. Vergne;
Organizing committee: N. Bergeron, S. Mehdi, M. Olbrich and N. Prudhon). Institut Henri Poincaré, Paris, France,
15-18 December 2008.

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 plan to give an exposition on the recent developments on the question about
the global natures of locally non-Riemannian homogeneous spaces,
with emphasis on the existence problem of compact forms, rigidity and deformation.

© Toshiyuki Kobayashi