On discontinuous group actions on non-Riemannian homogeneous spaces

نویسنده

  • Toshiyuki Kobayashi
چکیده

This article gives an up-to-date account of the theory of discrete group actions on non-Riemannian homogeneous spaces. As an introduction of the motifs of this article, we begin by reviewing the current knowledge of possible global forms of pseudoRiemannian manifolds with constant curvatures, and discuss what kind of problems we propose to pursue. For pseudo-Riemannian manifolds, isometric actions of discrete groups are not always properly discontinuous. The fundamental problem is to understand when discrete subgroups of Lie groups G act properly discontinuously on homogeneous spaces G/H for non-compact H. For this, we introduce the concepts from a group-theoretic perspective, including the ‘discontinuous dual’ of G/H that recovers H in a sense. We then summarize recent results giving criteria for the existence of properly discontinuous subgroups, and the known results and conjectures on the existence of cocompact ones. The final section discusses the deformation theory and in particular rigidity results for cocompact properly discontinuous groups for pseudo-Riemannian symmetric spaces. Translation from Sugaku, 57 (2005) 267–281, to appear in AMS translation journal “Sugaku Expositions”

برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید

ثبت نام

اگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید

منابع مشابه

6 On discontinuous group actions on non - Riemannian homogeneous spaces ∗

This article gives an up-to-date account of the theory of discrete group actions on non-Riemannian homogeneous spaces. As an introduction of the motifs of this article, we begin by reviewing the current knowledge of possible global forms of pseudoRiemannian manifolds with constant curvatures, and discuss what kind of problems we propose to pursue. For pseudo-Riemannian manifolds, isometric acti...

متن کامل

ACTION OF SEMISIMPLE ISOMERY GROUPS ON SOME RIEMANNIAN MANIFOLDS OF NONPOSITIVE CURVATURE

A manifold with a smooth action of a Lie group G is called G-manifold. In this paper we consider a complete Riemannian manifold M with the action of a closed and connected Lie subgroup G of the isometries. The dimension of the orbit space is called the cohomogeneity of the action. Manifolds having actions of cohomogeneity zero are called homogeneous. A classic theorem about Riemannian manifolds...

متن کامل

On 5-dimensional 2-step homogeneous randers nilmanifolds of Douglas type

‎In this paper we first obtain the non-Riemannian Randers metrics of Douglas type on two-step homogeneous nilmanifolds of dimension five‎. ‎Then we explicitly give the flag curvature formulae and the $S$-curvature formulae for the Randers metrics of Douglas type on these spaces‎. ‎Moreover‎, ‎we prove that the only simply connected five-dimensional two-step homogeneous Randers nilmanifolds of D...

متن کامل

2 3 Se p 20 05 Compact Clifford - Klein forms of symmetric spaces – revisited In memory of Professor Armand Borel

This article discusses the existence problem of a compact quotient of a symmetric space by a properly discontinuous group with emphasis on the non-Riemannian case. Discontinuous groups are not always abundant in a homogeneous space G/H if H is non-compact. The first half of the article elucidates general machinery to study discontinuous groups for G/H , followed by the most update and complete ...

متن کامل

Compact Clifford-Klein forms of symmetric spaces – revisited

This article discusses the existence problem of a compact quotient of a symmetric space by a properly discontinuous group with emphasis on the non-Riemannian case. Discontinuous groups are not always abundant in a homogeneous space G/H if H is non-compact. The first half of the article elucidates general machinery to study discontinuous groups for G/H, followed by the most update and complete l...

متن کامل

ذخیره در منابع من


  با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید

برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید

ثبت نام

اگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید

عنوان ژورنال:

دوره   شماره 

صفحات  -

تاریخ انتشار 2005