Hyperbolic Geometry from a Local Viewpoint: A Graduate Textbook on Two-Dimensional Hyperbolic Geometry, Metrics, and Fuc

Introduction

Hyperbolic geometry, a cornerstone of modern mathematics, finds its most elegant expression in the study of two-dimensional spaces. Hyperbolic Geometry from a Local Viewpoint by Linda Keen offers a rigorous yet accessible journey into this fascinating realm. Published by Cambridge University Press, this hardcover volume is tailored for graduate students and researchers who seek a deep, metric-first understanding of hyperbolic structures on plane domains. Written with clarity and precision, the book bridges classical concepts with contemporary applications, making it an indispensable resource for Indian students pursuing advanced studies in mathematics.

Book Overview

This book adopts a unique local approach: it begins with rigid motions in the Euclidean plane, using them as intuitive motivation to develop hyperbolic geometry within the unit disk. The metric is defined from an infinitesimal viewpoint—first establishing a density function, then integrating to obtain the global metric. From there, the narrative expands to arbitrary plane domains, introducing essential tools such as covering spaces, uniformization, and Fuchsian groups. The final chapters explore generalizations like hyperbolic and hyperbolic-like metrics, connecting to Kobayashi and Carathéodory metrics, and culminate in applications to holomorphic dynamics, complete with open problems for future research.

Key Highlights

• Infinitesimal Metric Construction: A step-by-step development from density to metric via integration.
• Comprehensive Coverage: From rigid motions and the unit disk to arbitrary plane domains and Fuchsian groups.
• Modern Generalizations: New material on hyperbolic-like metrics extending classical ideas.
• Applications to Dynamics: Holomorphic dynamics with accessible open problems for aspiring mathematicians.
• Graduate-Level Rigour: Written for students with a background in complex analysis and topology.

Inside the Book

The book is structured to build understanding layer by layer. Early chapters establish the hyperbolic metric on the unit disk, using the Poincaré disk model. Mid sections delve into covering spaces and uniformization, providing the language needed to study hyperbolic geometry on arbitrary domains. Later chapters introduce Fuchsian groups and their role in constructing quotient surfaces. The final part presents cutting-edge generalizations of hyperbolic metrics, linking them to complex analysis and dynamics. Each chapter includes exercises and examples that reinforce key concepts, making the book suitable for self-study or classroom use.

Key Topics

• Rigid motions and Euclidean geometry as a foundation
• Hyperbolic metric on the unit disk: density, geodesics, and isometries
• Covering spaces and the universal cover of plane domains
• Uniformization theorem and its implications
• Fuchsian groups and discontinuous actions
• Hyperbolic and hyperbolic-like metrics on arbitrary domains
• Kobayashi and Carathéodory metrics for plane domains
• Applications to holomorphic dynamics and open problems

Reader Benefits

By working through this book, readers gain a solid foundation in two-dimensional hyperbolic geometry from a fresh perspective. The local viewpoint simplifies complex global concepts, making them easier to grasp. The inclusion of modern generalizations prepares students for current research. Indian graduate students will appreciate the clear explanations and the focus on foundational ideas, which are essential for advanced work in geometry, topology, and dynamical systems. The open problems at the end offer a direct path to original research.

Learning Outcomes

• Understand the construction of hyperbolic metrics via infinitesimal densities.
• Analyze rigid motions and isometries in the hyperbolic plane.
• Apply covering space theory and uniformization to study hyperbolic structures.
• Work with Fuchsian groups and their fundamental domains.
• Generalize hyperbolic metrics to arbitrary plane domains.
• Connect hyperbolic geometry to holomorphic dynamics and explore open problems.

Who Should Read

This book is ideal for graduate students in mathematics, especially those specializing in geometry, complex analysis, or dynamical systems. Researchers seeking a modern treatment of hyperbolic metrics on plane domains will also find it valuable. Indian students preparing for competitive exams or pursuing PhDs in pure mathematics will benefit from its rigorous yet approachable style. Instructors looking for a textbook for a course on hyperbolic geometry will appreciate its clear structure and comprehensive coverage.

About the Author

Linda Keen is a distinguished mathematician known for her contributions to complex analysis, Teichmüller theory, and hyperbolic geometry. A professor emerita at the City University of New York, she has authored numerous influential papers and books. Her pedagogical expertise shines through in this volume, where she distills advanced concepts into digestible lessons for graduate students.

About the Publisher

Cambridge University Press is one of the world's oldest and most respected academic publishers. With a legacy spanning over four centuries, it is renowned for producing high-quality scholarly works in science, mathematics, and the humanities. This hardcover edition reflects the press's commitment to excellence, featuring durable binding and clear typesetting suitable for long-term study.

Conclusion

Hyperbolic Geometry from a Local Viewpoint is a masterful exposition that combines classical foundations with modern developments. Whether you are a graduate student embarking on research or a mathematician seeking to deepen your understanding, this book offers a rewarding intellectual journey. Order your copy from Bookshops.in today and explore the rich landscape of hyperbolic geometry through the lens of local metrics.