A Course in Metric Geometry by Dmitri Burago (PDF)

Description

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-Caratheodory 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 exposition.

User’s Reviews

Editorial Reviews: Review “The book is well worth reading. Contributing to this are the many elementary examples with which the authors supplement the text … Anyone who is intensely concerned with Riemmannian geometry will not pass up this book. It is so far without competition and fills a gap in the market.” —- Translated from Jahresbericht der Deutschen Mathematiker-Vereinigung

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐I adopted this text for a new graduate course that I taught on Metric Geometry.The other choice was `Spaces of Nonpositive Curvature’ by Bridson and Haefliger.I chose this text because it has exercises, it appears to cover nonpositive andnonnegative curvature in roughly equal proportion, and it’s less than half the price of Bridson-Haefliger.Unfortunately, the book has many, many typos. Fortunately, there is an errataconsisting of ten pages that can be found by googling “burago burago ivanov errata”.(Actually it’s more than just a list of typos: They correct a few false proofs as well).But beware! This list of errors is not complete.My students and I have found other typos as well. Of course, a well-trainedmathematician can get past the typos, but this text is supposed to be for first year graduate students.The writing is also rough in places. For example, polyhedral complexesare not rigorously defined. The style is Russian which I like, but it’snot the best example of this type of writing. The students have told me thatthey avoid reading the text whenever possible.The one redeeming feature may be the exercises. Some of the exercisesare excellent. Also the overall choice of topics is very good.With additional work this could be a great textbook, and (five years later) I am still hoping that theauthors will take the time to go through it at least a couple of more times.For now, I have switched to Bridson-Haefliger. I have abandoned this book.

⭐I am an undergraduate math student, and I encountered this book as the primary background text for a summer project in geometry. I had just been through a year or so of studying manifolds and basic Riemannian geometry, but it was my first substantial exposure to the length space approach to geometry as well as notions like curvature/Alexandrov spaces, polyhedral spaces, the hyperbolic plane, and quasi-isometries. It does a number of things very well, but it does possess some comparitively minor flaws that are nonetheless worth noting.Pros: The definitions and examples are very clearly explained and in a manner highly suitable for self-study. The exercises are challenging and they do a great deal to get the reader’s hands dirty and force him/her to really process the main ideas. Most of the proofs are very well written, and the authors do a good job of demonstrating the appropriate tools and outlining the best approaches in the proofs and the examples. The authors also do a good job in general of summoning the neccessary background material without dwelling on review too extensively; you should probably have already seen the Hausdorff measure and the fundamental group before you pick up this book, but the book recalls the main concepts and results (with helpful exercises) in case your memory has faded or don’t have much experience with such notions. Even the review of Riemannian geometry, which might be too long for some, is appropriate to firmly establish notation and because hyperbolic geometry and Riemannian curvature are so central to the topic of the text.Cons: Some of the key definitions and proofs (such as the triangle criterion for nonpositive/nonnegative curvature) contain typos, but for the most part they are sufficiently obvious that a critical reader will be able to figure out how to fix them with minimal confusion. While most of the proofs are easy to follow (especially the more technical proofs where lack of clarity would be a serious problem), some of the more visually-oriented proofs are very poorly written and there are precious few diagrams to compensate – see, for example, the example involving the curvature of cones over circles of various lengths. Also, the book could use some more exercises to accompany the more technical constructions. The exercises that it has are very well-chosen for the purpose of forcing the reader to internalize the relavent concepts and techniques, but some basic definition chasing to help process the harder definitions would make the process go smoother.

⭐The authors present a self-contained treatment of the geometry of length spaces. They begin with the definition of length spaces, and by the final chapter cover almost all of the material in the well known survey paper by Burago, Gromov, and Perel’man. Overall, the book provides an interesting and and accessible introduction to an important class of spaces (length spaces arise as limits of sequences of Riemannian manifolds). I found it exceedingly interesting that one can build analogues of most of the concepts from Riemannian geometry using only the intrinsic metric. There is great emphasis on exposition in this book; Burago makes great effort to motivate definitions and provide interesting examples. The reader is cautioned, however, that numerous typographic errors are to be found (some in fundamental definitions).My only complaint is the uneveness of the treatment. The authors spend two chapters developing some of the basic tools of Riemannian geometry in the setting of surfaces in R^3. It seems unlikely to me that a reader who is interested in length spaces would not already have at least passing aquaintence with Riemannian geometry. This, however, is a minor point, and I can recommend the book very highly.

⭐This book has only basic theory defined in good manner but to explain the theory through examples are very Vague. There is no balance between theory and examples that has been illustrated in it.

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