06181nam 22014535 450 991015474330332120190708092533.01-4008-8183-810.1515/9781400881833(CKB)3710000000631386(SSID)ssj0001651334(PQKBManifestationID)16426416(PQKBTitleCode)TC0001651334(PQKBWorkID)12555874(PQKB)10811084(MiAaPQ)EBC4738592(DE-B1597)467995(OCoLC)979579084(DE-B1597)9781400881833(EXLCZ)99371000000063138620190708d2016 fg engurcnu||||||||txtccrStrong Rigidity of Locally Symmetric Spaces. (AM-78), Volume 78 /G. Daniel MostowPrinceton, NJ : Princeton University Press, [2016]©19741 online resource (205 pages)Annals of Mathematics Studies ;247Bibliographic Level Mode of Issuance: Monograph0-691-08136-0 Includes bibliographical references.Frontmatter -- Contents -- §1. Introduction -- §2. Algebraic Preliminaries -- §3. The Geometry of χ : Preliminaries -- §4. A Metric Definition of the Maximal Boundary -- §5. Polar Parts -- §6. A Basic Inequality -- §7. Geometry of Neighboring Flats -- §8. Density Properties of Discrete Subgroups -- §8. Density Properties of Discrete Subgroups -- § 10. Pseudo Isometries of Simply Connected Spaces with Negative Curvature -- §11. Polar Regular Elements in Co-Compact Γ -- § 12. Pseudo-Isometric Invariance of Semi-Simple and Unipotent Elements -- §13. The Basic Approximation -- §14. The Map ∅̅ -- §15. The Boundary Map ∅0 -- §16. Tits Geometries -- §17. Rigidity for R-rank > 1 -- §18. The Restriction to Simple Groups -- §19. Spaces of R-rank 1 -- §20. The Boundary Semi-Metric -- §21. Quasi-Conformal Mappings Over K and Absolute Continuity on Almost All R-Circles -- §22. The Effect of Ergodicity -- §23. R-Rank 1 Rigidity Proof Concluded -- §24. Concluding Remarks -- Bibliography -- BackmatterLocally 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.Annals of mathematics studies ;Number 78.Riemannian manifoldsSymmetric spacesLie groupsRigidity (Geometry)Addition.Adjoint representation.Affine space.Approximation.Automorphism.Axiom.Big O notation.Boundary value problem.Cohomology.Compact Riemann surface.Compact space.Conjecture.Constant curvature.Corollary.Counterexample.Covering group.Covering space.Curvature.Diameter.Diffeomorphism.Differentiable function.Dimension.Direct product.Division algebra.Ergodicity.Erlangen program.Existence theorem.Exponential function.Finitely generated group.Fundamental domain.Fundamental group.Geometry.Half-space (geometry).Hausdorff distance.Hermitian matrix.Homeomorphism.Homomorphism.Hyperplane.Identity matrix.Inner automorphism.Isometry group.Jordan algebra.Matrix multiplication.Metric space.Morphism.Möbius transformation.Normal subgroup.Normalizing constant.Partially ordered set.Permutation.Projective space.Riemann surface.Riemannian geometry.Sectional curvature.Self-adjoint.Set function.Smoothness.Stereographic projection.Subgroup.Subset.Summation.Symmetric space.Tangent space.Tangent vector.Theorem.Topology.Tubular neighborhood.Two-dimensional space.Unit sphere.Vector group.Weyl group.Riemannian manifolds.Symmetric spaces.Lie groups.Rigidity (Geometry)516/.36Mostow G. Daniel, 606388DE-B1597DE-B1597BOOK9910154743303321Strong Rigidity of Locally Symmetric Spaces. (AM-78), Volume 782785740UNINA