The 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.
* The Schwarz Lemma and Its Generalizations * Volume Elements and the Schwarz Lemma * Distance and the Schwarz Lemma * Invariant Distances on Complex Manifolds * Holomorphic Mappings into Hyperbolic Manifolds * The Big Picard Theorem and Extension of Holomorphic Mappings * Generalization to Complex Spaces * Hyperbolic Manifolds and Minimal Models