Let G be a Lie group and H its closed subgroup. We say a discrete subgroup Γ of G is a discontinuous group for G/H if the natural action of Γ on G/H is properly discontinuous. If the action of Γ is furthermore fixed point free, then double coset space Γ G/H carries a natural manifold structure, which we say a Clifford-Klein form of a homogeneous manifold G/H. An important feature in our setting is that H is non-compact and that not all discrete subgroup of G can act properly discontinuously on G/H. Fundamental problems are:
Our concern is mainly with reductive cases and we shall present:
- Which homogeneous manifold G/H admits an infinite discontinuous group?
- Which homogeneous manifold G/H admits a compact Clifford-Klein form?
- A solution of the so called Calabi-Markus phenomenon.
- A necessary and sufficient condition for discrete subgroups to act properly discontinuously on homogeneous manifolds of reductive groups.
- A sufficient condition for the existence of compact Clifford-Klein forms.
- Conversely, a number of (explicitly computable) obstructions for the existence of compact Clifford-Klein forms of homogeneous manifolds.
© Toshiyuki Kobayashi