Metric Rigidity Theorems on Hermitian Locally Symmetric Manifolds (Pure Mathematics)

This monograph studies the problem of characterizing canonical metrics on Hermitian locally symmetric manifolds X of non-compact/compact types in terms of curvature conditions. The proofs of these metric rigidity theorems are applied to the study of holomorphic mappings between manifolds X of the same type. Moreover, a dual version of the generalized Frankel Conjecture on characterizing compact Kähler manifolds are also formulated. Front Cover......Page 1 Title......Page 4 Copyright......Page 5 Dedication......Page 6 TABLE OF CONTENTS......Page 10 PREFACE......Page 8 INTRODUCTION......Page 16 PART I BACKGROUND AND FIRST RESULTS......Page 22 1 Historical Background......Page 24 2 Statement of Results......Page 28 3 Deduction of Some Results from the Hermitian Metric Rigidity Theorem in the Seminegative Case......Page 30 1 Hermitian and Kahler Metrics......Page 32 2 The Hermitian Connection and its Curvature......Page 37 3 Different Notions of Positivity/Negativity of Curvature......Page 42 4 Projectivisation of Hermitian Holomcrphic Line Bundles......Page 49 1 Definition and Basic Properties of Riemannian Symmetric Manifolds......Page 55 2 Hermitian Symmetric Manifolds......Page 62 3 The Borel Embedding Theorem......Page 66 1 The Bergman and Carath?dory Metrics on Bounded Domains......Page 70 2 Classical Bounded Symmetric Domains......Page 76 3 Curvatures of Classical Bounded Symmetric Domains......Page 95 1 The Polydisc Theorem (and the Polysphere Theorem)......Page 103 2 The Harish-Chandra Embedding Theorem......Page 109 1 The Characteristic Bundle S......Page 114 2 An Integral Formula on S and an Algebraic Deduction of the Hermitian Metric Rigidity Theorem for Compact Quotients......Page 121 3 An Alternative Proof Using Moore's Ergodicity Theorem......Page 128 4 The Case of Irreducible and Locally Reducible Compact Quotients......Page 132 5 Applications of the Hermitian Metric Rigidity Theorem and Its Proofs......Page 137 1 Hermitian Symmetric Manifolds of Compact Type......Page 148 2 The Dual Characteristic Bundle S ?and an Integral Formula......Page 152 3 The Characteristic Bundle and Minimal Rational Curves......Page 157 4 Proof of the Metric Rigidity Theorem......Page 162 PART II FURTHER DEVELOPMENT......Page 172 1 Compactifications of Arithmetic Varieties and an Integral Formula......Page 174 2 An Alternative Proof in the K&hler Case......Page 180 1 The Equi-Diinensional Case......Page 188 2 Holomorphic Immersions Between Compact Hyperbolic Space Forms......Page 195 1 Homogeneous Hermitian Vector Bundle8 on Bounded Symmetric Domains......Page 209 2 An Extension of the Hermitian Metric Rigidity Theorem and Applications......Page 216 1 Formulation of the Problem......Page 228 2 Minimal Rational Curves on Hermitian Symmetric Manifolds of Compact Type......Page 231 3 Proof of the Rigidity Theorem for Holomorphic Mappings......Page 234 APPENDIX......Page 240 I.1 Semiaimple Lie Algebras ?General Theorems......Page 242 I.2 Cartan Subalgebras......Page 243 I.3 Semisimple Lie Algebras ?Structure Theory......Page 245 I.4 Representations of Semisimple Lie Algebras......Page 248 I.5 Some Results on Lie Groups and Their Representations......Page 252 II.2 Symmetric Manifolds......Page 254 III.1 Equivalent Definitions of Characteristic Vectors......Page 257 III.2 Characteristic Projective Subvarieties as Symmetric Projective Subinanifolds with Parallel Second Fundamental Forms......Page 260 III.3 Enumeration of the Characteristic Projective Subvarieties......Page 264 III.4 Higher Characteristic Bundles......Page 266 IV.1 Background......Page 269 IV.2 Formulation of a Dual Generalised Conjecture......Page 271 BIBLIOGRAPHY......Page 280 INDEX......Page 288 Pt. I. Background and first results. ch. 1. Historical background and summary of results -- ch. 2. Fundamentals of Hermitian and Kähler geometries -- ch. 3. Riemannian and Hermitian symmetric manifolds -- ch. 4. Bounded symmetric domains - the classical cases -- ch. 5. Bounded symmetric domains - general theory -- ch. 6. The Hermitian metric rigidity theorem for compact quotients -- ch. 7. The Kähler metric rigidity theorem in the semipositive case -- pt II. Further development -- ch. 8. The Hermitian metric rigidity theorem for quotients of finite volume -- ch. 9. The immersion problem for complex hyperbolic space forms -- ch. 10. The Hermitian metric rigidity theorem on locally homogeneous holomorphic vector bundles -- ch. 11 a rigidity theorem for holomorphic mappings between irreducible Hermitian symmetric manifolds of compact type PREFACE; TABLE OF CONTENTS; INTRODUCTION; PART I BACKGROUND AND FIRST RESULTS; CHAPTER 1 HISTORICAL BACKGROUND ANDSUMMARY OF RESULTS; 1 Historical Background; 2 Statement of Results; 3 Deduction of Some Results from the Hermitian Metric Rigidity Theorem in the Seminegative Case; CHAPTER 2 FUNDAMENTALS OF HERMITIAN AND KÄHLER GEOMETRIES; 1 Hermitian and Kahler Metrics; 2 The Hermitian Connection and its Curvature; 3 Different Notions of Positivity/Negativity of Curvature; 4 Projectivization of Hermitian Holomorphic Line Bundles; CHAPTER 3 RIEMANNIAN AND HERMITIAN SYMMETRIC MANIFOLDS Ngaiming Mok. Includes Bibliographical References (p. [265]-272) And Index.