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The only book that introduces differential geometry through a combination of an intuitive geometric foundation, a rigorous connection with the standard formalisms, computer exercises with Maple, and a problems-based approach.Starting with basic geometric ideas, Differential Geometry uses basic intuitive geometry as a starting point to make the material more accessible and the formalism more meaningful. The book presents topics through problems to provide readers with a deeper understanding. And, it introduces hyperbolic geometry in the first chapter rather than in a closing chapter as in other books.An important reference and resource book for any reader who needs to understand the foundations of differential geometry. ... Read more

Customer Reviews (1)

Fantastic concept, flawed execution
It's certainly a great concept: explain differential geometry - and andthe myriad real-world applications of this subject - by appealing to ourgeometric intuition! (Those who have read just about any other text willrealize that I am sincere, not sarcastic in this remark - the intuitiveapproach is quite unusual in treatments of the subject material.)

To alimited degree, the book is a success. The first chapter flows rathersmoothly, and could actually be used to introduce differential geometry inan advanced high school classroom. I would consider that in and over itselfto be a truimph! In places, it's fun to read, and some of the"constructions" (often using three dimensions) are both cleverand helpful. And I must confess that reading this book I picked up bits andpieces of intuition that I had missed when reading other texts.

For allof these reasons, I found myself really wanting to like this book; sadly, Iultimately found that I could not. Unfortunately, the intuitive approachstarts to break down as the book proceeds.In the later chapters, I couldonly intuitively grasp and fully understand what Henderson was trying toexplain because of previous familiarity with the material; I would havepretty baffled without prior knowledge of the subject. The writing andpresentation just does not compare with that in some of the better (if moretraditional) texts in differential geometry, such as Manfredo P. Do Carmo'sDiffertial Geometry of Curves and Surfaces or Michael Spivak's excellentfive-volume Comprehensive Introduction to Differential Geometry. If one isfamiliar with those (or other similar) texts, it might be fun to take alook at Henderson's book. If not, look there first - or at least look thereas well - in your explorations of this field of mathematics. ... Read more

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This is the third version of a book on Differential Manifolds; in this latest expansion three chapters have been added on Riemannian and pseudo-Riemannian geometry, and the section on sprays and Stokes' theorem have been rewritten. This text provides an introduction to basic concepts in differential topology, differential geometry and differential equations. In differential topology one studies classes of maps and the possibility of finding differentiable maps in them, and one uses differentiable structures on manifolds to determine their topological structure. In differential geometry one adds structures to the manifold (vector fields, sprays, a metric, and so forth) and studies their properties. In differential equations one studies vector fields and their integral curves, singular points, stable and unstable manifolds, and the like. ... Read more

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Modern, but....
This book is a proper subset of Lang's later book "Fundamentals of Differential Geometry (Graduate Texts in Mathematics, 191)".

Not a "first book", ok as reference
Lang's book is definitely not useful as textbook for classes or for self-guided study (learnt this the hard way). He is rather abstract and provides zero motivation for the theory. The book is obviously made for people who learnt diff. geometry elsewhere but want to read a cleaner and more modern treatment. To this end, Lang's book is useful. The best part is that manifolds are infinite-dimensional right away. This is probably the only reason for buying Lang instead of/in addition to Dieudonne as a reference. Otherwise, the book is a little too terse; fiber bundles are merely hinted at. Moreover, I think some of the proofs are unnecessarily complicated, such as the one for Frobenius theorem.

Maybe Lang's best book.
Well, we have here another book on differential manifolds, and another book by Serge Lang. Lang is well-known by writing (lots of) books on different topics in analysis and algebra, all of them in a quite "Bourbaki-like" style: attaining maximum generality, with less motivation than most students would like. This is no surprise, because Lang himself is a Bourbakist.

So, what's interesting about D&RM? It's a book very much like Lang's other books, only that here the Bourbakist's approach is quite happy: it's one of the very few books on his subject to present most of his results in infinite-dimensional(Banach) version, a must if you are interested in nonlinear functional analysis or dynamical systems. The exposition is very clean and clear: Lang uses categories all the way to estabilish the main relations between the different differential-topological structures and tools, and he does not hesitate in stating and using tools from analysis, such as Lebesgue measure and functional analysis' main theorems. The proofs are very polished and, in a certain sense, beautiful, a philosophy thatpermeates most of the book. As if it weren't enough, the book still contains an appendix with a Von Neumann's seminar about the spectral theorem.

All things considered, it's a quite "state-of-the-art" book about the basics of differential manifolds, from an analyst's perspective. This perspective provides differential topology with a lot of additional clarity and power. I don't know if most physicists would like this book, because its motivations, if any, are sparse and sometimes quite obscure, as long as physical applications are concerned. For a mathematician, however, this book is a gem: it's Lang at its best, and the perfect opening door to global analysis (the nonlinear analysis on infinite dimensional manifolds, a vast field of mathematics that encompasses dynamical systems and nonlinear functional analysis). Despite all that, I would also recommend to physicists to at least tackle this book, as an antidote to all the crap that the so-called "differential topology for physicists" books put on their heads, because I don't know a cleaner and more precise presentation of differential manifolds so far. ... Read more

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Quantum groups and quantum algebras as well as non-commutative differential geometry are important in mathematics and considered to be useful tools for model building in statistical and quantum physics. This book, addressing scientists and postgraduates, contains a detailed and rather complete presentation of the algebraic framework. Introductory chapters deal with background material such as Lie and Hopf superalgebras, Lie super-bialgebras, or formal power series. Great care was taken to present a reliable collection of formulae and to unify the notation, making this volume a useful work of reference for mathematicians and mathematical physicists. ... Read more

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This book collects invited contributions by specialists in the domain of elliptic partial differential equations and geometric flows. The articles provide a balance between introductory surveys and the most recent research, with a unique perspective on singular phenomena. Notions such as scans and the study of the evolution by curvature of networks of curves are completely new and lead the reader to the frontiers of the domain.

The intended readership are postgraduate students and researchers in the fields of elliptic and parabolic partial differential equations that arise from variational problems, as well as researchers in related fields such as particle physics and optimization.

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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

Customer 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 fullydiscussed there by L.P. Eisenhart were even rediscovered decades later ---and published another time as new results. This book is, of course, writtenin the old good traditional style, one will not find here, e.g., Cartan'sforms, but it is really an everlasting treasure. Look also for theContinuous Groups of Transformations by the same author. ... Read more

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Customer Reviews (3)

Captain Obvious Handles Another Review
This book contains a lot of information about manifolds, particularly those with differentiable structures. It used to only be available with a boring green cover and it was expensive. Now, its cover is colorful and has a wacky picture on it. I thought that this would surely make the price go up but it got cheaper! The picture on the front cover concerns operations on manofolds, particularly differentiable manifolds.

Almost everything about higher-dim smooth manfolds in 220 pgs
Don't be deceived by the title of Kosinski's "Differential Manifolds," which sounds like a book covering differential forms, such as Lee's Introduction to Smooth Manifolds, or by claims that it is self-contained or for beginning graduate students. In fact, the purpose of this book is to lay out the theory of (higher-dimensional, i.e., >= 5) smooth manifolds as it was known in the '60s, namely, the techniques of handle decompositions, framed cobordism including the Thom-Pontrjagin construction, and surgery (sometimes called spherical modification). Offhand, I can't think of another book that covers all these topics as thoroughly and concisely, and does so in a way that is readily comprehensible.

The first 4 chapters are an overview of the basic background of differential topology - differential manifolds, diffeomorphism, imbeddings and immersions, isotopy, normal bundles, tubular neighborhoods, Morse functions, intersection numbers, transversality - as one would find in, e.g., Guillemin and Pollack's Differential Topology, Milnor's Topology from the Differentiable Viewpoint, or Hirsch's Differential Topology, albeit at a higher level and with much less explanation. As the author himself states, with some understatement, "The presentation is complete, but it is assumed, implicitly, that the subject is not totally unfamiliar to the reader." Although I would dispute somewhat the notion that it is complete, as several very important results on immersions and isotopies of Whitney and Haefliger are cited and used repeatedly, but not proved, since, as the author explains, it would have taken the reader too far a field. The reader should also have a good knowledge of algebraic topology (Dold and Spanier are frequently used as references), as well as the classification of bundles over spheres as found in Steenrod.

Since the purpose of the first 4 chapters (about 75 pp) is to develop the machinery of differential topology to the point where the results on handles, cobordism, and surgery can be proved, several topics are briefly touched upon that are generally not encountered in introductory diff top books, such as the group Gamma of differential structures on the m-sphere mod those that extend over the m-disk or the bidegree of a map from a product of spheres to a sphere, in addition to the aforementioned results of Whitney and Haefliger, but just enough is given so that they may be used in later proofs. Most perplexing is Chapter V, on foliations, which has only a tenuous connection to the preceding material and absolutely none to the following. It seems that the author just included it because he felt that knowledge of the subject was essential for a topologist, not because it was necessary for the purposes of this book; it certainly could be skipped, but is worth reading as a brief introduction to foliations.

The heart of the book is Chapter VI, where the concept of gluing manifolds together is explored. Normally, connected sums are defined by removing imbedded balls in 2 closed manifolds and gluing them along the spherical boundaries, but Kosinski instead constructs, explicitly in local coordinates, an orientation-reversing diffeomorphism of a punctured ball and then uses that to identify punctured balls in each manifold. Similarly, handle attachment is defined, rather than by just attaching a handle to an imbedded sphere in the boundary, but instead by again explicitly constructing an orientation reversing diffeomorphism of a (in the 0-dim case) punctured hemisphere and then identifying it with the normal bundle of a point in the boundary of the manifold. In this way, one automatically constructs smooth manifolds without having to resort to "vigorous hand waving" to smooth corners. The downside to this method (which is likely to be unfamiliar to modern readers) is that much time is spent constructing explicit formulas for handle attachments, e.g., in local coordinates, but after Chapter VI the details of these maps are no longer needed.

The last 4 chapters are the most interesting, as all the tools developed in the first 6 chapters are used to prove results such as the existence of handle decompositions for manifolds; the classification of handlebodies; the h-cobordism theorem, proved much easier than in Milnor's Lectures on the h-Cobordism Theorem; the Poincare conjecture for dimensions > 4; Poincare duality (for smooth manifolds only); the Morse inequalities; the existence of Heegard diagrams; the equivalence of the aforementioned group Gamma with the group of differential structures on the sphere and with h-cobordism classes of homotopy spheres (Theta); the Pontrjagin-Thom isomorphism; results on stably parallelizable and almost parallelizable manifolds; conditions under which surgery can eliminate homology in the middle dimension of a framed manifold that is closed or has a boundary that is a homotopy sphere, thus leading to corollaries about when a manifold is cobordant to a highly-connected manifold (such as a sphere); and the computation of some of the aforementioned Theta groups. As you can see, a lot of important results are derived, whose proofs are complete except for a few technical lemmas that are cited.

Most chapters conclude with a section titled "Historical Remarks" or just "Remarks," that explains the history of the development of the subject, including many references. The author himself, now almost 80, had in hand in some of these developments and was personally well-familiar with the giants of 20-c. mathematics who discovered them, such as Thom, Bott, Milnor, Smale, Whitney, Wall, Browder, Morse, etc. The text is also interlaced with exercises, most of which are relatively straightforward.

The book concludes with a new appendix, written last year by John Morgan (my former thesis adviser), on Perelman's proof of the Poincare conjecture. It's just an overview of the proof and feels really out of place, the only connections being that it concerns the Poincare conjecture in dim 3, whose proof for dimensions higher than 4 is one of the highlights of this book, and also that Perelman's proof involves a kind of surgery. This appendix does little to enhance the value of the book.

The book is not without it faults, however. In addition to the above observations about it being too advanced for an introductory text and the incongruity of Chapter V, there are the usual batch of typos: an arrow pointing the wrong way in a diagram on pg 231; a wrong sign in the second displayed equation on pg 102; the switching between indices 0,1 and 1,2 on pg 93; the reversal of the equations for the equator and meridian, as well as the words themselves, on pg. 212; 1/2 in place of epsilon 3 lines above eqn (2.2.6) on pg 128; missing bars over the h in many places in pp. 110-11, as well as omitting the -1 exponent for g in one place; etc. There are also errors of exposition, such as reversing the order of the i and j terms in the definition of M1 and M2 on pg. 211, which leads to factors of +/- missing from subsequent formulas, that fortunately do not impact the results, but do waste the reader's time; this category would also include Case 2 on pg. 214, whose proof is identical to that for Case 1 after a framed surgery and thus unnecessary, or even the 2 possibilities for m, listed in the first sentence for both Case 1 and Case 2 on that page, that are in fact identical, as well as an extraneous condition on n on pg 171. A more serious omission is Theorem X,5.1 (in the notation of the book), which should have been stated in 2 ways, one of which being analogous to Theorems X,4.5 and X,3.4 for use in proving the corollaries 5.2 and 5.3.

Probably the worst mistake is when the term "framed manifold" is introduced and defined to mean exactly the same thing as "pi-manifold," without ever acknowledging this fact, and then the terms are used interchangeably afterward, with theorems about framed manifolds being proved by reference to results about pi-manifolds, and even with the redundant expression "framed pi-manifold" being used in a few places. Moreover, "framed cobordant" is then defined in Chapter X to mean something different than it meant in Chapter IX.

Another group of complaints that I have is with the system of references. First of all, the chapter numbers do not appear in either the running heads or the theorem numbers, so when a result is cited in a previous chapter, the reader must flip back and forth through the book to find it, remembering the chapter numbers for each chapter, or must go back to the table of contents to locate it. Moreover, many theorems from earlier chapters are used without comment, or a reference is made to a theorem when in fact a corollary is being used (or vice versa!). Sometimes a theorem from another source is cited as the justification for a statement, when in fact the author is directly applying a theorem from his own book that just happened to use that other author's result in its proof - citing his own theorem, by number, would save the reader a lot of effort. And then there's the important imbedding theorem of Haefliger that he frequently cites, even though he never actually states what the theorem says! (I had to read Haefliger's paper to verify that it actually could be used to produce the results that Kosinski wanted.)

Rigorous but not inaccessible.
This book treats differential topology from scratch to the works by Pontryagin, Thom, Milnor, and Smale. The best thing with this book is that you don't too often have to read between the lines. That is, the exposition is detailed and user-friendly, so I highly recommend this book. If something is to be blamed, the price is horrendous. ... Read more

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This graduate-level monographic textbook treats applied differential geometry from a modern scientific perspective. Co-authored by the originator of the world s leading human motion simulatorHuman Biodynamics Engine , a complex, 264-DOF bio-mechanical system, modeled by differential-geometric tools this is the first book that combines modern differential geometry with a wide spectrum of applications, from modern mechanics and physics, via nonlinear control, to biology and human sciences. The book is designed for a two-semester course, which gives mathematicians a variety of applications for their theory and physicists, as well as other scientists and engineers, a strong theory underlying their models. ... Read more

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Differential geometry studies geometrical objects using analytical methods. Like modern analysis itself, differential geometry originates in classical mechanics. For instance, geodesics and minimal surfaces are defined via variational principles and the curvature of a curve is easily interpreted as the acceleration with respect to the path length parameter. Modern differential geometry in its turn strongly contributed to modern physics. This book gives an introduction to the basics of differential geometry, keeping in mind the natural origin of many geometrical quantities, as well as the applications of differential geometry and its methods to other sciences. The text is divided into three parts. The first part covers the basics of curves and surfaces, while the second part is designed as an introduction to smooth manifolds and Riemannian geometry. In particular, Chapter 5 contains short introductions to hyperbolic geometry and geometrical principles of special relativity theory. Here, only a basic knowledge of algebra, calculus and ordinary differential equations is required. The third part is more advanced and introduces into matrix Lie groups and Lie algebras the representation theory of groups, symplectic and Poisson geometry, and applications of complex analysis in surface theory. The book is based on lectures the author held regularly at Novosibirsk State University. It is addressed to students as well as anyone who wants to learn the basics of differential geometry. ... Read more

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This self-contained textbook presents an exposition of the well-known classical two-dimensional geometries, such as Euclidean, spherical, hyperbolic, and the locally Euclidean torus, and introduces the basic concepts of Euler numbers for topological triangulations, and Riemannian metrics. The careful discussion of these classical examples provides students with an introduction to the more general theory of curved spaces developed later in the book, as represented by embedded surfaces in Euclidean 3-space, and their generalization to abstract surfaces equipped with Riemannian metrics. Themes running throughout include those of geodesic curves, polygonal approximations to triangulations, Gaussian curvature, and the link to topology provided by the Gauss-Bonnet theorem. Numerous diagrams help bring the key points to life and helpful examples and exercises are included to aid understanding. Throughout the emphasis is placed on explicit proofs, making this text ideal for any student with a basic background in analysis and algebra. ... Read more

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Using a self-contained and concise treatment of modern differential geometry, this book will be of great interest to graduate students and researchers in applied mathematics or theoretical physics working in field theory, particle physics, or general relativity.The authors begin with an elementary presentation of differential forms.This formalism is then used to discuss physical examples, followed by a generalization of the mathematics and physics presented to manifolds. The book emphasizes the applications of differential geometry concerned with gauge theories in particle physics and general relativity.Topics discussed include Yang-Mills theories, gravity, fiber bundles, monopoles, instantons, spinors, and anomalies. ... Read more

Customer Reviews (3)

NOTHING NEW
THIS BOOK IS JUST ANOTHER EXAMPLE OF THE MANY PUBLICATIONS IN THIS SUBJECT MATTER THAT MISSES THE WHOLE POINT.

Concise, big picture treatment of the subject
This text, while lacking in rigour and detail, is an ideal supplement for self-study or lectures on modern mathematical methods in physics.In fact, it is precisely its lack of detail that allows it to act as the yin to the yang of other, weightier texts.Most books on this subject obscure the big picture behind their equations, reducing pleasant geometry to the grimy level of analysis.No such crime is committed here, and the reader is much the better for it.To be sure, this is not a stand-alone text - to not delve into the details would only enter the reader into the false security of ignorance.However, it is most definitely a must-have book for anyone interested in modern physics and mathematics.

Recommended texts to accompany this one are: 1) Geometry of Physics, Frankel 2) Intro to Lie Algebras & Rep. Th., Humphreys 3) Geometry, Topology,& Physics, Nakahara (another useful survey) 4) Spin Geometry, Lawson & Michelson

excellent introduction to relevant topics!
This is a concise introduction to applications of differential geometry on some improtant topics in physics, such as gauge theories, gravity...etc. Despite its size (which is rather comfortable for readers who prefer lessabstract definitions and theorems), nothing essential to the spirit of thetopics has been missed, I personally think it sure is one of the excellentbooks on these subjects and am glad to recommand it to you all who love andwant to discover the geometric aspects of physics! ... Read more

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Explores the interplay between geometry and topology in the simplest nontrivial case: the surfaces of constant curvature. DLC: Surfaces of constant curvature. ... Read more

Customer Reviews (4)

passionnant
Ce livre est à la fois une très bonne introduction à différentes idées mathématiques, comme l'espace hyperbolique ou la topologie algébrique... et un bon compléments à des cours de géométrie différentielle ou théorie des groupes...
On le dévore facilement grâce à son style entrainent et ces très bonnes illustrations.

Excellent book
An excellent exposition of the three geometries of surfaces. I highly recommend this book to advanced undergraduates and beginning graduates in mathematics.

Geometry from an isometry group point of view
The three basic geometries of constant curvature are the Euclidean (zero curvature), spherical (positive curvature) and hyperbolic (negative curvature). These may be studied through their isometries (chapters 1, 3, 4, respectively). This is pretty. Other than these three "planes" one may obtain surfaces that are locally isometric to them by taking quotients by certain groups of isometries. That's easy in the Euclidean case (chapter 2), trivial in the spherical case, and hard in the hyperbolic case (chapter 5), which needs to be complemented by a whole chapter of topology (chapter 6). These groups of isometries have "fundamental regions", i.e. polygons that tessellate the plane in which they live. Not all tessellations are obtained in this way, however, so one is lead to study tesellations in general, corresponding to more general groups of isometries (chapters 7-8). The presentation is well motivated within its own aesthetic framework, but some discomfort results from the feeling that this clever approach is a virtuoso post-construction (some may say that this is consistent with "Klein's spirit" (cf. p. viii)). Historical background and related topics are treated in informal discussion sections at the end of each chapter. The main theme here is the deep connections with complex function theory.

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

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This established reference work continues to lead its readers to some of the hottest topics of contemporary mathematical research. This new edition introduces and explains the ideas of the parabolic methods that have recently found such spectacular success in the work of Perelman at the examples of closed geodesics and harmonic forms. It also discusses further examples of geometric variational problems from quantum field theory, another source of profound new ideas and methods in geometry.

Customer Reviews (2)

maths background for General Relativity and QFT
For theoretical physicists, especially those studying Einstein's Theory of General Relativity, Or if your subject is quantum field theory. Jost's book is good preparation. He offers an in-depth teaching of Riemannian geometry. So ideas like covariant and contravariant derivatives on a manifold take on elegant meaning.

Note that General Relativity does not get an explicit mention. However, a typical physics GR course might often not have time to give a good discussion of the underlying maths. And standard GR texts, like Misner, Thorne and Wheeler or Weinberg, also tend to have very abbreviated explanations of the maths. So Jost's book is useful for those of you inclined to look further.

The length of the book means it's probably too long for a standard 1 term or semester course, if the intent is to entirely cover the book.

Intro to Riemannian Geom. and Geom. Analysis
Covers standard material on Reimannian Geometry. In addition: variational problems from QFT. Spin geometry and Dirac operators are explained in detail. ... Read more

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Integral geometry originated with problems on geometrical probability and convex bodies. Its later developments have proved to be useful in several fields ranging from pure mathematics (measure theory, continuous groups) to technical and applied disciplines (pattern recognition, stereology). The book is a systematic exposition of the theory and a compilation of the main results in the field. The volume can be used to complement courses on differential geometry, Lie groups, or probability or differential geometry. It is ideal both as a reference and for those wishing to enter the field. ... Read more