A Course in Metric Geometry

Preface This book is not a research monograph or a reference book (although research interests of the authors influenced it a lot)—this is a textbook. Its structure is similar to that of a graduate course. A graduate course usually begins with a course description, and so do we. Course description. The objective of this book is twofold. First of all, we wanted to give a detailed exposition of basic notions and techniques in the theory of length spaces, a theory which experienced a very fast development in the past few decades and penetrated into many other mathematical disciplines (such as Group Theory, Dynamical Systems, and Partial Differential Equations). However, we have a wider goal of giving an elementary introduction into a broad variety of the most geometrical topics in geometry—the ones related to the notion of distance. This is the reason why we included metric introductions to Riemannian and hyperbolic geometries. This book tends to work with " easy-to-touch " mathematical objects by means of " easy-to-visualize " methods. There is a remarkable book [Gro3], which gives a vast panorama of " geometrical mathematics from a metric viewpoint ". Unfortunately , Gromov's book seems hardly accessible to graduate students and non-experts in geometry. One of the objectives of this book is to bridge the gap between students and researchers interested in metric geometry, and modern mathematical literature. Prerequisite. It is minimal. We set a challenging goal of making the core part of the book accessible to first-year graduate students. Our expectations of the reader's background gradually grow as we move further in the book. We tried to introduce and illustrate most of new concepts and methods by using their simplest case and avoiding technicalities that take attention xi xii Preface away from the gist of the matter. For instance, our introduction to Riemann-ian geometry begins with metrics on planar regions, and we even avoid the notion of a manifold. Of course, manifolds do show up in more advanced sections. Some exercises and remarks assume more mathematical background than the rest of our exposition; they are optional, and a reader unfamiliar with some notions can just ignore them. For instance, solid background in differential geometry of curves and surfaces in R 3 is not a mandatory prerequisite for this book. However, we would hope that the reader possesses some knowledge of differential geometry, and from time to time we draw analogies …