Book Description

Differential geometry has become a standard tool in the analysis of statistical models, offering a deeper appreciation of existing methodologies and highlighting the issues that can be hidden in an algebraic development of a problem. This volume is the first to apply these techniques to econometrics. An introductory chapter provides a brief tutorial for those unfamiliar with the tools of differential geometry. The following chapters offer applications of geometric methods to practical solutions and offer insight into problems of econometric inference. ... Read more

Book Description

Topology of Surfaces, Knots, and Manifolds offers an intuition-based and example-driven approach to the basic ideas and problems involving manifolds, particularly one- and two-dimensional manifolds. A blend of examples and exercises leads the reader to anticipate general definitions and theorems concerning curves, surfaces, knots, and links--the objects of interest in the appealing set of mathematical ideas known as "rubber sheet geometry." The result is a book that provides solid coverage of the mathematics underlying these topics. ... Read more

Reviews (3)

Fun, but not very substantive.
This is a relatively fun romp through some very interesting concepts, but it lacks rigor.The book could have been much stronger if the author had simply developed some of the basic concepts (compactness, connectedness, homeomorphisms, homotopy, etc) rather than do a little hand-waving around a nice illustration.As it stands, this book is only 140 pages long, and does not develop any of its topics (manifolds, surfaces, graphs, knots) adequately.This book is far too weak to serve as a good text.Kinsey's TOPOLOGY OF SURFACES is much stronger, and costs less.Or look as something like Gamelin's INTRO TO TOPOLOGY.Or even Schaum's outline GENERAL TOPOLOGY, which deals with the basics, but is highly readable and rigorous.

Very Misleading Title, Quite Thin, No Rigor, and Overpriced
This book is subtitled "A First Undergraduate Course" but is certainly below undergraduate level.A high school student could easily follow this--which might be a good thing in certain cases--but the rigor is lacking.In fact, there is barely a hint of any rigor whatsoever.It is mostly intuitive arguments and the author often says things like "but we won't bother worrying about mathematical technicalities".It does get you to be able to visualize certain things well, but the visualization techniques can be found in other books also.The book is very thin and a quick read--hardly worth the money they are trying to get for it.If you're really at the undergraduate level and want to learn some topology, try something like Mendelson's "Introduction to Topology" by Dover or one of the excellent topology books in the series "Undergraduate Texts in Mathematics" by Springer.Munkres is also a classic.If you're not an undergraduate in a math related field and just want to know about the ideas behind topology or perhaps see some visualization techniques, try something like "The Shape of Space" by Weeks.Overall I was very disappointed with this text.If you could purchase this book for under $20 it might be worth it, but even then I think the other books I quoted are better in both price and substance.

No math library is complete without this book
This book presents the topology of surfaces, manifolds and knots in a manner that is reachable for undergraduate students with only a knowledge of calculus.Some linear algebra might be helpful.The text is written in a style that is easy to follow and there are superfluous examples.The exercises in the text are well thought out and are not extremely difficult.The exercises complement the text very well.The text makes clear a lot of difficult concepts such as isotopic surfaces as opposed to homeomorphic surfaces.I particularly enjoyed the manner in which the topology of knots was explained.After reading this text, the reader should be able to better visualize the projective plain and even the Klein bottle as it exists in 4-dimensional space.I have not read a text on topology that I enjoyed reading as much since Munkres.This text is a must have for any topologist. ... Read more

Book Description

This invaluable book, based on the many years of teaching experience of both authors, introduces the reader to the basic ideas in differential topology. Among the topics covered are smooth manifolds and maps, the structure of the tangent bundle and its associates, the calculation of real cohomology groups using differential forms (de Rham theory), and applications such as the Poincaré–Hopf theorem relating the Euler number of a manifold and the index of a vector field. Each chapter contains exercises of varying difficulty for which solutions are provided. Special features include examples drawn from geometric manifolds in dimension 3 and Brieskorn varieties in dimensions 5 and 7, as well as detailed calculations for the cohomology groups of spheres and tori. ... Read more

Reviews (1)

This book Rules!
This book is just so full of useful information and details. It has a lot of problems for which most of the solutions are supplied. Man, I love differential manifolds after spending some quality time with this book. ... Read more

Book Description

Excellent brief introduction presents fundamental theory of curves and surfaces and applies them to a number of examples. Topics include curves, theory of surfaces, fundamental equations, geometry on a surface, envelopes, conformal mapping, minimal surfaces, more. Well-illustrated, with abundant problems and solutions. Bibliography.
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Reviews (2)

Good treatment of classical differential geometry
Struik's book provides solid coverage of curve and surface theory from the classical point of view, i.e. the kind of stuff Monge, Serret, Frenet and Gauss did. I agree that the book should be on the shelves of mathematicians. A number of classical topics are simply not in vogue these days, and one can find them discussed at length in Struik, or in the exercises. In this sense the book certainly has a more geometric flavor than a number of contemporary texts.

However, Struik can't be used to understand what is happening today. For these purposes,books by O'Neill and do Carmo would be more appropriate. The discussion of manifolds and coordinate charts, the discussion of connection forms, differential forms, covariant derivatives, exterior derivatives, pullbacks and pushforwards can be found in these texts. This is the language of modern geometry.It leads on naturally to tensors, fibre bundles, de Rham cohomology and so on and so forth.The emphasis in modern geometry is on global phenomena, the interaction between local and global (e.g. Morse theory or De Rham cohomology), and the attempt to do everything in an algebraic setting (projective modules, spectral sequences, categories etc.) For this purpose, Struik is useless, though he does have some coverage of forms (he calls them by their earlier name of 'pfaffians').

The price of the book makes it an attractive purchase.

Struik's book - a classic on classical differential geometry
I simply cannot believe I am the first reviewer of this book!This book should be on the shelf of every mathematician interested in geometry, every computer graphics specialist, everyone interested in solid modelling. For ten bucks, you get a great summary of a wide range of topics in "classical differential geometry" -- the stuff geometers were interested in one hundred years ago. Today it's gauge and string theory -- but the topics discussed in this book are timeless, and many have seen remarkable renaissances in recent years. It is a wonderful little book ... I am using it to teach a basic differential geometry course next year. ... Read more

Book Description

Recent developments in the field of differential geometry have been so extensive that a new book with particular emphasis on current work in Riemannian geometry is clearly necessary.This new text brilliantly serves that purpose and includes an elementary account of twistor spaces that will interest both applied mathematicians and physicists. It presents recent developments in the theory of harmonic spaces, commutative spaces, and mean-value theorems previously only available in the source literature.The final chapter provides the only account available in book form of manifolds known as `Willmore surfaces', illustrated by a series of computer-generated pictures. This book is sure to be welcomed by researchers, teachers, and students interested in the latest developments in differential geometry. ... Read more

Reviews (1)

A Good Book!
This book is one of the most advanced text on Riemannian manifold that I have ever read. Especially the last two chapers which cover recent active topics in differential geometry, we can find only in research papers. ... Read more

Book Description

In his classic work of geometry, Euclid focused on the properties of flat surfaces. In the age of exploration, mapmakers such as Mercator had to concern themselves with the properties of spherical surfaces. The study of curved surfaces, or non-Euclidean geometry, flowered in the late nineteenth century, as mathematicians such as Riemann increasingly questioned Euclid's parallel postulate, and by relaxing this constraint derived a wealth of new results. These seemingly abstract properties found immediate application in physics upon Einstein's introduction of the general theory of relativity.

In this book, Eisenhart succinctly surveys the key concepts of Riemannian geometry, addressing mathematicians and theoretical physicists alike. ... Read more

Reviews (1)

The best classical-style exposition of Riemannian Geometry.
I bought the Russian translation of this book in 1954 and found that this is the best source of the Riemannian geometry, not only for a beginner (as I was at that time), but also for every specialist. Some items fully discussed there by L.P. Eisenhart were even rediscovered decades later --- and published another time as new results. This book is, of course, written in the old good traditional style, one will not find here, e.g., Cartan's forms, but it is really an everlasting treasure. Look also for the Continuous Groups of Transformations by the same author. ... Read more

Book Description

Prerequisite: solid understanding of basic theory of finite dimensional vector spaces and their linear transformations, point-set topology and advanced calculus. Last four chapters investigate some of the consequences which the existence of a differentiable structure entails in various situations. 1963 ed.
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Book Description

The geometry of the hyperbolic plane has been an active and fascinating field of mathematical inquiry for most of the past two centuries. This book provides a self-contained introduction to the subject, taking the approach that hyperbolic geometry consists of the study of those quantities invariant under the action of a natural group of transformations. Topics covered include the upper half-space model of the hyperbolic plane, Möbius transformations, the general Möbius group and the subgroup preserving path length in the upper half-space model, arc-length and distance, the Poincaré disc model, convex subsets of the hyperbolic plane, and the Gauss-Bonnet formula for the area of a hyperbolic polygon and its applications. The style and level of the book, which assumes few mathematical prerequisites, make it an ideal introduction to this subject, providing the reader with a firm grasp of the concepts and techniques of this beautiful area of mathematics. ... Read more

Reviews (1)

great book
this is a really great introduction to hyperbolic geometry.especially if you want to study gammas acting on the upper half plane.it starts at a much lower level then any other text. ... Read more

Book Description

This volume brings together lectures from a conference on spectral theory and geometry held under the auspices of the International Centre for Mathematical Sciences in Edinburgh. The contributions by world experts include expanded versions of many of the original lectures. Together, they survey the core material and go beyond to reach deeper results. For graduate students and experts alike, this book will be a highly useful resource. ... Read more

Book Description

The author uses modern methods from computational group theory and representation theory to treat this classical topic of function theory. He provides classifications of all automorphism groups up to genus 48. The book also classifies the ordinary characters for several groups, arising from the action of automorphisms on the space of holomorphic abelian differentials of a compact Reimann surface. This book is suitable for graduate students and researchers in group theory, representation theory, complex analysis and computer algebra. ... Read more

Book Description

In Euclidean geometry, constructions are made with ruler and compass. Projective geometry is simpler: its constructions require only a ruler. In projective geometry one never measures anything, instead, one relates one set of points to another by a projectivity. The first two chapters of this book introduce the important concepts of the subject and provide the logical foundations. The third and fourth chapters introduce the famous theorems of Desargues and Pappus. Chapters 5 and 6 make use of projectivities on a line and plane, repectively. The next three chapters develop a self-contained account of von Staudt's approach to the theory of conics. The modern approach used in that development is exploited in chapter 10, which deals with the simplest finite geometry that is rich enough to illustrate all the theorems nontrivially. The concluding chapters shows the connections among projective, Euclidean, and analytic geometry. ... Read more

Book Description

This book is an introduction to the theory of complex manifolds. The authors¿ intent is to familiarize the reader with the most important branches and methods in complex analysis of several variables and to do this as simply as possible. Therefore, the abstract concepts involving sheaves, coherence, and higher-dimensional cohomology have been completely avoided. Only elementary methods such as power series, holomorphic vector bundles, and one-dimensional cocycles are used. Nevertheless, deep results can be proved. The book can be used as a first introduction to several complex variables as well as a reference for the expert. ... Read more

Book Description

This book offers a new treatment of the topic, one which is designed to make differential geometry an approachable subject for advanced undergraduates. Professor McCleary considers the historical development of non-Euclidean geometry, placing differential geometry in the context of geometry students will be familiar with from high school. The text serves as both an introduction to the classical differential geometry of curves and surfaces and as a history of a particular surface, the non-Euclidean or hyperbolic plane. The main theorems of non-Euclidean geometry are presented along with their historical development.The author then introduces the methods of differential geometry and develops them toward the goal of constructing models of the hyperbolic plane.While interesting diversions are offered, such as Huygen's pendulum clock and mathematical cartography, the book thoroughly treats the models of non-Euclidean geometry and the modern ideas of abstract surfaces and manifolds. ... Read more

Reviews (2)

a great book!
This is a great book.The author develops the differential geometry of curves and surfaces.The endpoint is the vindication of Euclid's parallel postulate.I thoroughly enjoyed reading this book.Very readable.

Excellent text connecting classical to differential geometry
This book is ideal for those with a long time interest in mathematics or the student just becoming interested in advanced topics.It successfully takes the concepts of classic geometry (Euclidean), clearly explains how the parallel postulate interacts with the other postulates and then introduces differential geometry as a natural outgrowth of hyperbolic geometry.McLeary's book succeeds by demonstrating the connection of modern differential geometry to the concepts in which we were educated.This is not a book for the casual reader, but includes many problems and solutions to the more interesting of them ... Read more

Book Description

The Second Edition combines a traditional approach with the symbolic manipulation abilities of Mathematica to explain and develop the classical theory of curves and surfaces. You will learn to reproduce and study interesting curves and surfaces - many more than are included in typical texts - using computer methods. By plotting geometric objects and studying the printed result, teachers and students can understand concepts geometrically and see the effect of changes in parameters.Modern Differential Geometry of Curves and Surfaces with Mathematicaexplains how to define and compute standard geometric functions, for example the curvature of curves, and presents a dialect of Mathematica for constructing new curves and surfaces from old. The book also explores how to apply techniques from analysis.Although the book makes extensive use of Mathematica, readers without access to that program can perform the calculations in the text by hand. While single- and multi-variable calculus, some linear algebra, and a few concepts of point set topology are needed to understand the theory, no computer or Mathematica skills are required to understand the concepts presented in the text. In fact, it serves as an excellent introduction to Mathematica, and includes fully documented programs written for use with Mathematica.Ideal for both classroom use and self-study, Modern Differential Geometry of Curves and Surfaces with Mathematicahas been tested extensively in the classroom and used in professional short courses throughout the world. ... Read more

Reviews (6)

No CD-Rom!
The main purpose of this textbook is supposed to be the combination of the visualization of curves and surfaces along with the traditional differential geometry materials. This book makes a great effort to realize this lofty goal with some success as a reference book. As a main textbook, however, it fails to deliver what it promised by overlooking one small point: the lack of CD-Rom. The readers are expected to type in the sample programs manually. This can be very time-consuming especially for inexperienced students. Inevitably, a lot of valuable class time has be consumed helping students looking for and correcting errors, many of which are small typographic errors completely unrelated to either their mathematical understanding or their computer skill.

Instructors can attempt writing their own Mathematica class-notes, but copyright issue will come up if they don't be careful distributing the notebook files to the students. All such hassle can easily be eliminated with the CD-Rom(s). Hopefully this issue is resolved when the next edition is printed. Until then, I cannot recommend this book as a main textbook.

Good introduction to differential geometry
The visualization of complicated geometrical objects
is now routine thanks to the excellent software that
has been developed over the past two decades. Now
students and professionals can have a better
appreciation of the geometrical properties of these
objects thanks to these software packages. In this
book the author has done a great job of doing this,
having chosen one of the best tools for this purpose:
Mathematica. The book is a hefty one, totaling almost
1100 pages, but its perusal is worth the effort for
those who want a more intuitive appreciation behind
the concepts of differential geometry. Physicists in
particular, who usually need a pictorial approach to
complement the learning of a subject, should really
enjoy this book. It could definitely be used as a
textbook in a beginning course in differential
geometry since there are problems at the end of each
chapter and most of the results in the book are proven
with the required mathematical rigor, I.e. this book
is not just code and pictures, and a substantial
portion of it is devoted to definitions and rigorous
proofs. This is especially true for the discussion on
differentiable manifolds and Riemannian geometry. The
author also includes a brief biography of the
mathematicians who have been involved in differential
geometry at various places in the book. The
Mathematica code in the book though can be revised to
make it look more like standard mathematical notation,
thanks to the new features of Mathematica that have
appeared since this book was published (1997). The use
of color shading is not done in the book, except for a
short insert with pictures of several surfaces, but
the reader can easily experiment with the color
functions available in Mathematica if needed. A very
lengthy appendix that lists the functions and code
used in the book is included.
Some of the concepts that are usually
difficult to grasp intuitively for those approaching
differential geometry for the first time but are here
illustrated nicely include: 1. The computation of the
curvature of plane curves and the plotting of this
curvature. The curvature of the famous Lissajous
curves, very familiar from oscilloscope traces, is
computed. The author might have spent a little more
time explaining why the curvature plots have the shape
they do however. 2. The treatment of osculating curves
to plane curves. 3. The finding of curves whose
curvature is equal to the arc length times a Bessel
function. The resulting plots are very entertaining.
4. The computation of the torsion of a curve in space.
The discussion on torus knots is particularly well-
done. 5. The author's discussion on surfaces in
Euclidean space motivates well the concept of a
differentiable manifold. He plots a few surfaces with
coordinate patches that have a singularity, and shows
how to plot surfaces that defined nonparametrically.

Kummer's surface, of particular importance in
algebraic geometry, is plotted here. Even more useful
is the author's treatment of nonorientable surfaces,
wherein he shows the reader how to plot the Moebius
strip, the Klein bottle, and two realizations of the
projective plane using Mathematica. Several examples
of the Gaussian curvature of surfaces are plotted. The
Gauss map, one of the most important tools for the
physicist, is given detailed treatment. 6. Rare in
textbooks at this level of differential geometry is a
discussion of minimal surfaces, but the author gives a
very nice treatment in this book. The Enneper's,
Scherk's Henneberg's and Catalan's minimal surfaces
are plotted along with the Gauss map of Enneper's
surface. Minimal surfaces are extremely important in
theoretical physics, such as superstring and membrane
theories, and are also very important in optimization
theory, so it was nice to see a discussion of them
included in the book. In recent years galleries of
minimal surfaces have appeared on the Web, and this
book allows one to plot these without too much effort.
The author even introduces the use of complex analysis
in the study of minimal surfaces. Readers interested
in understanding the mathematics of string theory will
appreciate this discussion. In addition, the
Weierstrass representation, which allows generation of
new minimal surfaces, is introduced. Readers familiar
with the Weierstrass function for elliptic curves will
see it used here for this generation.

Great Book!
Gray does not intend for you to buy his book if you don't haveaccess to Mathematica and simply want to learn about differentialgeometry from an axiomatic standpoint. Of course if you don't have access to Mathematica, this isn't for you, and even if you do have Mathematica, you will probably want to have a good "standard" text to go along with your learning. Having said this, the book and Mathematica make an excellent addition to anyone's diferential geometry course.

Excellent overall book
I strongly disagree with the reviewer at the bottom of this page. Having taken a differential geometry course last year using do Carmo's book (also excellent) I came to appreciate the intuition that this book lends to the reader. Also, this book makes greater use of elementary linear algebra than is common in some more standard texts, for example in defining the second fundamental form in terms of the Shape Operator. For students wanting to compliment their course notes or standard text with a book which will thoroughly explain both the fundamentals and isolated topics, this book is highly recommended.

Impressive both in size and content
I would recommend this book to anyone that needs an intuitive introduction to the subject that is complete in many ways and that provides visualization and examples using mathematica when needed. Before purchasing this I was expecting the treatment to be 'informal' and using mathematica rather than mathematical rigour to introduce concepts and results. This is not the case however. In my opinion the author has struck the right balance between a formal maths treatment and the abilities provided by mathematica to make the book easier to read and coprehend. ... Read more

Reviews (1)

Deserves 10 stars
This book should have been titled "A Hilbert Space Idea/Problem Book" as it not only challenges the reader to work out interesting problems in operator theory and the geometry of Hilbert space, but also motivates the essential ideas behind these fields. It is definitely a book that, even though out-of-print, will be referred to by many newcomers to operator theory and quantum physics. The insight one gains by the reading of this book is unequaled in any other books in existence on operator theory. It is becoming more rare as mathematics advances, to find books that attempt to explain the intuition behind the abstractions that are manifested in any area of mathematics. The problems in the book deal with both concrete examples and general theorems, and the reader should attempt to try and solve them without looking at the hints. The solutions found by the reader can then be compared with the author's, and some interesting differences will occur.

There are so many interesting discussions in this book that to list them all would probably entail listing everything in the book. The reader will find excellent discussions of the origin of normal operators on infinite dimensional Hilbert spaces as analogs to matrices on finite dimensional spaces; why the weak topology in infinite dimensions is not metrizable; the non-emptiness of the spectrum and why the spectral radius can be computed even though the spectrum cannot; the impossibility of isolated singular operators; the non-continuity of the spectrum: the existence of an operator with a large spectrum and the existence of operators with small spectra in every neighborhood of the large spectrum. The author then goes on to show that the spectrum is an upper semicontinuous function, thus preventing the existence of small spectra arbitrarily close to large spectra. This is an excellent discussion on the meaning and intuition behind semicontinuity; the result that every normal operator is unitarily equivalent to a multiplication and its equivalance to the spectral theorem. The author goes on to explain how one gives up the sigma-finiteness of the measure when doing this, and the origin of functional calculus; the difference between infinite and finite dimensions when attempting a polar decomposition for operators and its connection with partial isometries; the origin of compact operators and their connection with integral equations. The author shows how even the identity operator is not an integral operator on the space of square-integrable functions with Lebesgue measure.

In discussing the spectral theorem in chapter 13 the author statesmost profoundly: "In some contexts some authors choose to avoid a proof that uses the spectral theorem even if the alternative is longer and more involved. This sort of ritual circumlocution is common to many parts of mathematics; it is the fate of many big theorems to be more honored in evasion than in use. The reason is not just mathematical mischievousness. Often a long but 'elementary' proof gives more insight, and leads to more fruitful generalizations, than a short proof whose brevity is made possible by a powerful but overly specialized tool." In these few sentences the author has characterized the problem with current methods of teaching advanced mathematics. Too often the formalism masks the true meaning and intuitive motivation behind the mathematics. And even though mathematics is being applied to many different areas at an unprecedented rate, pure mathematics seems to be trapped in a local minimum, and I beleive this is due to the reluctance of authors to explain in detail the essentials of their ideas. This book is a perfect example of how mathematics can be taught that requires much thought and creativity on the part of students, without spoon-feeding them and thus encouraging a passive attitude to the learning of mathematics. I salute the author in his achievements in research and in teaching...one can only hope that his approach will be followed in all future works of mathematics. ... Read more

Book Description

Cartan geometries were the first examples of connections on a principal bundle. They seem to be almost unknown these days, in spite of the great beauty and conceptual power they confer on geometry. The aim of the present book is to fill the gap in the literature on differential geometry by the missing notion of Cartan connections. Although the author had in mind a book accessible to graduate students, potential readers would also include working differential geometers who would like to know more about what Cartan did, which was to give a notion of "espaces généralisés" (= Cartan geometries) generalizing homogeneous spaces (= Klein geometries) in the same way that Riemannian geometry generalizes Euclidean geometry. In addition, physicists will be interested to see the fully satisfying way in which their gauge theory can be truly regarded as geometry. ... Read more

Reviews (1)

Everything via principal bundles.
Sharpe's book is a detailed argument supporting the assertion that most of differential geometry can be considered the study of principal bundles and connections on them, disguised as an introductory differential geometrytextbook.

Some standard introductory material (e.g. Stokes' theorem) isomitted, as Sharpe confesses in his preface, but otherwise this is a trulywonderful place to read about the central role of Lie groups, principalbundles, and connections in differential geometry.The theme is that whatone can do for Lie groups, one can do fiberwise for principal bundles, toyield information about the base.

The informal style (just look at thetable of contents) and wealth of classical examples make this book apleasure to read.While its somewhat nonstandard approach and preferencefor classical terminology might confuse those who have never beenintroduced to the concepts, this is a perfect *second* place to read andmarvel about differential geometry. ... Read more

Book Description

This text intends to provide the student with the knowledge of a geometry of greater scope than the classical geometry taught today, which is no longer an adequate basis for mathematics or physics, both of which are becoming increasingly geometric. The geometry of surfaces is an ideal starting point for students learning geometry for the following reasons; first, the extensions offer the simplest possible introduction to fundamentals of modern geometry: curvature, group actions and covering spaces. Second, the prerequisites are modest and standard and include only a little linear algebra, calculus, basic group theory and basic topology. Third and most important, the theory of surfaces of constant curvature has maximal connectivity with the rest of mathematics. They realize all the topological types of compact two-dimensional manifolds, and historically, they are the source of the main concepts of complex analysis, differential geometry, topology, and combinatorial group theory, as well as such hot topics as fractal geometry and string theory. The formal coverage is extended by exercises and informal discussions throughout the text. ... Read more

Reviews (1)

Interesting advanced undergraduate course
Stillwell contends (in his preface) that the geometry of surfaces of constant curvature is an ideal topic for such a course, and he gives three convincing reasons for that, the most important one being "maximal connectivity with the rest of mathematics," which he elucidates. I applaud this.

He then demurs that such a deep and broad topic cannot be covered completely by a book of his modest size. He does include, at the end of each chapter, informal discussions of further results and references to the literature - these are very valuable.

The teacher of the teacher of Stillwell's teacher was Felix Klein, and Stillwell approaches his subject in the spirit of Klein. His first chapter describes in detail the group of isometries of the Euclidean plane E. Then his second chapter gives the Hopf-Killing classification of complete, connected Euclidean surfaces as quotient spaces of E by certain groups of isometries of E, and up to isometry there are exactly five such (cylinder, twisted cylinder, torus, Klein bottle and E itself). The proof introduces the student to the important subject of covering spaces.

Stillwell's writing style is pleasantly informal but can be careless. The main subject of the book is surfaces, but he never defines "surface!" He does define the compound "Euclidean surface," but his definition is inadequate: he doesn't require that his distance function only take on positive real values for distinct points, and he doesn't specify the conditions that it be a metric (e.g., triangle inequality). Evidently a Euclidean surface is a metric space that is locally isometric to E.

The next two chapters are very good introductions to two-dimensional spherical, elliptic and hyperbolic geometries, again with a description of their isometries. The hyperbolic plane is introduced by first showing nicely that the pseudosphere has Gaussian curvature -1, and then transferring a suitable coordinate system and infinitesimal distance function on the pseudosphere over to the upper half-plane H.

Stillwell asserts without proof that Gaussian curvature is well-defined (for "surfaces" in Euclidean three-space); he gives no reference for that result. He does not mention Gauss' Theorema Egregrium either. In fact he pretty much skirts differential geometry altogether in this book.

The meat of the book is chapter 5 on hyperbolic surfaces (metric spaces which are locally isometric to H). He states without proof Rado's theorem that any compact surface is homeomorphic to the identification space of a polygon (he doesn't explain that "surface" in this theorem means two-dimensional topological manifold). He applies this result to show that such surfaces can be "realized geometrically". He doesn't define that either, but from his argument we glean that such topological surfaces can underly a structure of either Euclidean, hyperbolic or spherical surface (locally isometric to the sphere S).

Chapter 6 begins with the classification of compact topological surfaces and their fundamental groups. For a "geometric surface" X, which now means a quotient of either E, H or S by a discontinuous fixed-point-free group G of isometries, he proves that G is isomorphic to the fundamental group of X. He is able to define a "geodesic path" on X without using differential geometry, but warns of difficulties with "geodesic monogons." He proves that on a compact orientable surface of genus > 1, each non-trivial free homotopy class has a unique geodesic representative.

The final two chapters are a nice treatment of tessellations.

In sum, this book is a very good introduction for advanced undergraduates to the portion of surface geometry that interests Stillwell. It is an attractive mixture of topology, algebra and a smidgen of analysis. ... Read more

Book Description

The book contains a clear exposition of two contemporary topics in modern differential geometry: - distance geometric analysis on manifolds, in particular, comparison theory for distance functions in spaces which have well defined bounds on their curvature- the application of the Lichnerowicz formula for Dirac operators to the study of Gromov's invariants to measure the K-theoretic size of a Riemannian manifold.It is intended for both graduate students and researchers who want to get a quick and modern introduction to these topics. ... Read more