*Existence Problem of Compact Locally Symmetric Spaces*,

Journées
Solstice d'été 2007: Théorie de Lie,
Géométrie et Représentations (organized by B. Keller,
B. Klingler, R. Rentschler and O. Schiffmann), Institut de
Mathématiques de Jussieu, Paris, France, 21-23 June 2007.

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.

**References**

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