Strong Rigidity of Locally Symmetric Spaces

Front Cover
Princeton University Press, 1973 - Mathematics - 195 pages
0 Reviews

Locally symmetric spaces are generalizations of spaces of constant curvature. In this book the author presents the proof of a remarkable phenomenon, which he calls "strong rigidity": this is a stronger form of the deformation rigidity that has been investigated by Selberg, Calabi-Vesentini, Weil, Borel, and Raghunathan.

The proof combines the theory of semi-simple Lie groups, discrete subgroups, the geometry of E. Cartan's symmetric Riemannian spaces, elements of ergodic theory, and the fundamental theorem of projective geometry as applied to Tit's geometries. In his proof the author introduces two new notions having independent interest: one is "pseudo-isometries"; the other is a notion of a quasi-conformal mapping over the division algebra K (K equals real, complex, quaternion, or Cayley numbers). The author attempts to make the account accessible to readers with diverse backgrounds, and the book contains capsule descriptions of the various theories that enter the proof.

  

What people are saying - Write a review

We haven't found any reviews in the usual places.

Contents

1 Introduction
3
2 Algebraic Preliminaries
10
Preliminaries
20
A Metric Definition of the Maximal Boundary
31
5 Polar Parts
35
6 A Basic Inequality
44
7 Geometry of Neighboring Flats
52
8 Density Properties of Discrete Subgroups
62
14 The Map 0
103
15 The Boundary Map 0
107
16 Tits Geometry
120
17 RRank Greater than One
125
18 Reduction to Simple Groups
128
19 Spaces of RRank 1
134
20 The Boundary SemiMetric
142
21 QuasiConformal Mappings Over K and Absolute Continuity on Almost All RCircles
156

9 PseudoIsometries
66
10 PseudoIsometries of Simply Connected Spaces with Negative Curvature
71
11 Polar Regular Elements in CoCompact V
76
12 PseudoIsometric Invariance of SemiSimple and Unipotent Elements
80
13 The Basic Approximation
96
22 The Effect of Ergodicity
169
23 RRank 1 Rigidity Proof Concluded
180
Wt Concluding Remarks
187
Bibliography
193
Copyright

Common terms and phrases

References to this book

All Book Search results »

Bibliographic information