Metric Rigidity Theorems on Hermitian Locally Symmetric Manifolds

The following sections are included:

• PREFACE

• TABLE OF CONTENTS

• INTRODUCTION

The following sections are included:

• Historical Background

• Statement of Results

• Deduction of Some Results from the Hermitian Metric Rigidity Theorem in the Seminegative Case

The following sections are included:

• Hermitian and Kähler Metrics

• The Hermitian Connection and its Curvature

• Different Notions of Positivity/Negativity of Curvature

• Projectivization of Hermitian Holomorphic Line Bundles

The following sections are included:

• Definition and Basic Properties of Riemannian Symmetric Manifolds

• Hermitian Symmetric Manifolds

• The Borel Embedding Theorem

The following sections are included:

• The Bergman and Carathéodory Metrics on Bounded Domains

• Classical Bounded Symmetric Domains

• Curvatures of Classical Bounded Symmetric Domains

The following sections are included:

• The Polydisc Theorem (and the Polysphere Theorem)

• The Harish–Chandra Embedding Theorem

The following sections are included:

• The Characteristic Bundle S

• An Integral Formula on S and an Algebraic Deduction of the Hermitian Metric Rigidity Theorem for Compact Quotients

• An Alternative Proof Using Moore's Ergodicity Theorem

• The Case of Irreducible and Locally Reducible Compact Quotients

• Applications of the Hermitian Metric Rigidity Theorem and Its Proofs

The following sections are included:

• Hermitian Symmetric Manifolds of Compact Type

• The Dual Characteristic Bundle S* and an Integral Formula

• The Characteristic Bundle and Minimal Rational Curves

• Proof of the Metric Rigidity Theorem

The following sections are included:

• Compactifications of Arithmetic Varieties and an Integral Formula

• An Alternative Proof in the Kähler Case

The following sections are included:

• The Equi–dimensional Case

• Holomorphic Immersions Between Compact Hyperbolic Space Forms

The following sections are included:

• Homogeneous Hermitian Vector Bundles on Bounded Symmetric Domains

• An Extension of the Hermitian Metric Rigidity Theorem and Applications

The following sections are included:

• Formulation of the Problem

• Minimal Rational Curves on Hermitian Symmetric Manifolds of Compact Type

• Proof of the Rigidity Theorem for Holomorphic Mappings

The following sections are included:

• SEMISIMPLE LIE ALGEBRAS AND THEIR REPRESENTATIONS
  • Semisimple Lie Algebras – General Theorems
  • Cartan Subalgebras
  • Semisimple Lie Algebras – Structure Theory
  • Representations of Semisimple Lie Algebras
  • Some Results on Lie groups and Their Representations

• SOME THEOREMS IN RIEMANNIAN GEOMETRY
  • The de Rham Decomposition Theorem
  • Some Theorems on Riemannian Locally Symmetric Manifolds

• CHARACTERISITIC PROJECTIVE SUBVARIETIES ASSOCIATED TO HERMITIAN SYMMETRIC MANIFOLDS
  • Equivalent Definitions of Characteristic Vectors
  • Characteristic Projective Subvarieties as Symmetric Projective Sub– manifolds with Parallel Second Fundamental Forms
  • Enumeration of the Characteristic Projective Subvarieties
  • Higher Characteristic Bundles

• A DUAL GENERALIZED FRANKEL CONJECTURE FOR COMPACT KÄHLER MANIFOLDS OF SEMINEGATIVE BISECTIONAL CURVATURE
  • Background
  • Formulation of a Dual Generalized Frankel Conjecture

• BIBLIOGRAPHY

• INDEX