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 various other kinds of geometry (pseudo-Riemannian, 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 highlight complex symmetric structures, and discuss some of the results of Teduka who discussed when sufrace groups arise as the fundamental groups of manifolds which are locally complex symmetric.

Our key idea is the criterion for proper actions, and the study of the action of outer automorphisms on nilpotent orbits.

© Toshiyuki Kobayashi