## Hyperbolic Manifolds and Holomorphic Mappings: An IntroductionThe first edition of this influential book, published in 1970, opened up a completely new field of invariant metrics and hyperbolic manifolds. The large number of papers on the topics covered by the book written since its appearance led Mathematical Reviews to create two new subsections ?invariant metrics and pseudo-distances? and ?hyperbolic complex manifolds? within the section ?holomorphic mappings?. The invariant distance introduced in the first edition is now called the ?Kobayashi distance?, and the hyperbolicity in the sense of this book is called the ?Kobayashi hyperbolicity? to distinguish it from other hyperbolicities. This book continues to serve as the best introduction to hyperbolic complex analysis and geometry and is easily accessible to students since very little is assumed. The new edition adds comments on the most recent developments in the field. |

### Contents

Chapter I The Schwarz Lemma and Its Generalizations | 1 |

Chapter II Volume Elements and the Schwarz Lemma | 17 |

Chapter IV Invariant Distances on Complex Manifolds | 45 |

Chapter V Holomorphic Mappings into Hyperbolic Manifolds | 67 |

Chapter VI The Big Picard Theorem and Extension of Holomorphic Mappings | 77 |

Chapter VII Generalization to Complex Spaces | 93 |

Chapter VIII Hyperbolic Manifolds and Minimal Models | 103 |

Chapter IX Miscellany | 115 |

Postscript | 129 |

Bibliography | 135 |

Summary of Notations | 143 |

145 | |

147 | |

### Other editions - View all

Hyperbolic Manifolds and Holomorphic Mappings: An Introduction Shoshichi Kobayashi No preview available - 2005 |

### Common terms and phrases

assertion assume automorphism ball biholomorphic boundary bounded domain called Carath´eodory distance Chapter choose closed coincides compact compact complex manifold complete hyperbolic complex manifold complex space connected consider construction contained converges coordinate system Corollary covering curvature curve defined deﬁnite definition denote dimension distance distance-decreasing ds2D ds2M equivalent everywhere Example exists extended fact ﬁrst follows give given Hence Hermitian holomorphic functions holomorphic mapping holomorphic transformations homogeneous hyperbolic manifold identity implies inequality integer invariant Kobayashi line bundle linear locally mapping f Math measure metric minimal n-dimensional neighborhood normal obtain origin positive number Problem projection Proof Proposition prove pseudodistance punctured disk respect result Ricci tensor satisfying sequence similar singular submanifold subsequence suitable Theorem 3.1 trivial unit disk vector volume element zero