## 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. |

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### 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 |

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

analytic subset assume automorphism Bergman kernel Bergman metric big Picard theorem biholomorphic Carath´eodory distance Chapter Chern compact complex manifold compact Kaehler manifold compact subset complete hyperbolic complete with respect complex manifold complex space complex submanifold converges coordinate system Corollary curvature is bounded deﬁned denote dim+ distance-decreasing domain in Cn ds2D Example extended ﬁnite ﬁrst Gaussian curvature group of holomorphic Hence Hermitian manifold holomorphic functions holomorphic mapping holomorphic sectional curvature holomorphic transformations homogeneous hyperbolic complex space hyperbolic manifold hypersurface inequality integer intentionally left blank Kaehler manifold Kobayashi Let f line bundle mapping f meromorphic mapping metric ds2M minimal model morphic n-dimensional negative constant negative deﬁnite neighborhood obtain open unit disk Pn(C polydisk positive number Proof Let proof of Theorem prove punctured disk relatively minimal Ricci tensor satisﬁes Schwarz lemma sectional curvature Siegel domain Theorem 3.1 trivial vector bundle zero