A Course in Metric Geometry
About this Title
Dmitri Burago, Pennsylvania State University, University Park, PA, Yuri Burago, Steklov Institute of Mathematics, St. Petersburg, Russia and Sergei Ivanov, Steklov Institute of Mathematics, St. Petersburg, Russia
Publication: Graduate Studies in Mathematics
Publication Year 2001: Volume 33
ISBNs: 978-0-8218-2129-9 (print); 978-1-4704-1794-9 (online)
MathSciNet review: MR1835418
MSC: Primary 53C23
“Metric geometry” is an approach to geometry based on the notion of length on a topological space. This approach experienced a very fast development in the last few decades and penetrated into many other mathematical disciplines, such as group theory, dynamical systems, and partial differential equations.
The objective of this graduate textbook is twofold: to give a detailed exposition of basic notions and techniques used in the theory of length spaces, and, more generally, to offer an elementary introduction into a broad variety of geometrical topics related to the notion of distance, including Riemannian and Carnot-Carathéodory metrics, the hyperbolic plane, distance-volume inequalities, asymptotic geometry (large scale, coarse), Gromov hyperbolic spaces, convergence of metric spaces, and Alexandrov spaces (non-positively and non-negatively curved spaces). The authors tend to work with “easy-to-touch” mathematical objects using “easy-to-visualize” methods.
The authors set a challenging goal of making the core parts of the book accessible to first-year graduate students. Most new concepts and methods are introduced and illustrated using simplest cases and avoiding technicalities. The book contains many exercises, which form a vital part of the exposition.
Advanced undergraduates, graduate students, and research mathematicians interested in geometry and specialists in related fields.
Table of Contents
- Chapter 1. Metric Spaces
- Chapter 2. Length Spaces
- Chapter 3. Constructions
- Chapter 4. Spaces of Bounded Curvature
- Chapter 5. Smooth Length Structures
- Chapter 6. Curvature of Riemannian Metrics
- Chapter 7. Space of Metric Spaces
- Chapter 8. Large-scale Geometry
- Chapter 9. Spaces of Curvature Bounded Above
- Chapter 10. Spaces of Curvature Bounded Below