Product Description
This book provides a unified combinatorial realization of the categroies of (closed, oriented) 3-manifolds, combed 3-manifolds, framed 3-manifolds and spin 3-manifolds. In all four cases the objects of the realization are finite enhanced graphs, and only finitely many local moves have to be taken into account. These realizations are based on the notion of branched standard spine, introduced in the book as a combination of the notion of branched surface with that of standard spine. The book is intended for readers interested in low-dimensional topology, and some familiarity with the basics is assumed. A list of questions, some of which concerning relations with the theory of quantum invariants, is enclosed. ... Read more

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The past two decades have brought explosive growth in 4-manifold theory. Many books are currently appearing that approach the topic from viewpoints such as gauge theory or algebraic geometry. This volume, however, offers an exposition from a topological point of view. It bridges the gap to other disciplines and presents classical but important topological techniques that have not previously appeared in the literature. Part I of the text presents the basics of the theory at the second-year graduate level and offers an overview of current research. Part II is devoted to an exposition of Kirby calculus, or handlebody theory on 4-manifolds. It is both elementary and comprehensive. Part III offers in depth a broad range of topics from current 4-manifold research. Topics include branched coverings and the geography of complex surfaces, elliptic and Lefschetz fibrations, $h$-cobordisms, symplectic 4-manifolds, and Stein surfaces. Applications are featured, and there are over 300 illustrations and numerous exercises with solutions in the book. ... Read more

Customer Reviews (3)

Review

Actually I was looking for loose ends - things that do not appear in Scorpan's _The Wild World of 4-Manifolds_, like the Buzaca construction of exotic R4s, or the construction of an ``universal'' R4, and I found it it Gompf's book.

(In fact I'm interested in exotic forcing-generic R4s and their import, if any, in General Relativity. Truly wild beasts...)


Francisco Antonio Doria

Extremely detailed overview of Kirby calculus
Readers familiar with the proof of Stephen Smale's proof of the high-dimensional Poincare conjecture will know that handle calculus was employed in the proof. This book is an overview of Kirby calculus, which is essentially handle calculus in dimensions less than or equal to four.

Kirby calculus can be used to describe four-dimensional manifolds such as elliptic surfaces, and gives a pictorial description of its handle decomposition. Its utility lies further than this however, as Kirby calculus has been used to answer questions that would have been very difficult otherwise.

The book begins with a very quick overview of the algebraic topology and gauge theory of four-dimensional manifolds. Readers not familiar with this material will have to consult other books or papers on the subject.

Part two takes up Kirby calculus, and handle decompositions are described with examples given for disk bundles over surfaces and tori. Handle moves are employed as processes that allow one to go from one description of a manifold to another. Handlebody descriptions are given for spin manifolds, and more exotic topics, such as Casson handles and branched covers are treated.

Part 3 of the book uses techniques from algebraic geometry to describe branched covers of algebraic surfaces. Handle decompositions of Lefschetz fibrations are given, and its is shown that a Stein structure on a manifold is completely described by a handle diagram. There is also a thorough discussion of exotic structures on Euclidean 4-space. In spite of the non-constructive nature of these results, namely that no explicit example of an exotic structure is given, the discussion is a fascinating one and has recently been shown to be important in physics.

The reader will no doubt attempt many of the exercises; the solutions of some of these given in the back of the book. The book serves well the needs of those dedicated individuals who are interested in specializing in low-dimensional topology. In addition, physicists interested in these ideas couuld benefit from its reading, although some of the results may seem a little heavy-handed and abtruse at times.

Complexity is the name...
If you really into mathematics, this book is for you. It contains comprehensive explanation of the Kirby calculas. The complexity of this book require graduate level mathematics knowleadge as a prerequisite. Itdescribes in the detail of a closed 4-manifold which admits a finitedecomposition into geometric pieces of finite volume. It also consider thehomotopy types of closed 4-manifolds which are Seifert fibred or which arethe total spaces of bundles with base and fibre closed aspherical surfaces. ... Read more

Product Description
An introduction to basic ideas in differential topology, based on the many years of teaching experience of both authors. 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 Poincare-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

Customer Reviews (2)

A mile wide and a yard deep
Barden & Thomas's "Introduction to Differential Manifolds" has the broadest coverage of any introductory graduate text in differential topology that I've seen, even more than Lee's Introduction to Smooth Manifolds or Guillemin & Pollack's Differential Topology, and in less than 200 pages. Not only does it cover the standard topics found in all such books, i.e., the rank theorem, diffeomorphisms, immersions, embeddings, tangent bundles, Sard's theorem, the Whitney embedding theorem, etc.; more topological topics, such as degree theory, the Poincare-Hopf theorem, Morse theory, and handlebodies; and the usual material with a more analytic or geometric flavor, such as differential forms, tensors, vector bundles, integration, Stokes's theorem, de Rham cohomology, and Lie groups and algebras, but there is also a chapter on fibre bundles (which already is rare for a book of this level) that includes further material on classifications of higher-dimensional manifolds that probably has never appeared in book-form before. Unfortunately, the treatment of many of these topics is rather cursory, with the most interesting material being the least well explained.

Except for Chapter 3 and the latter parts of Chapter 7, the book should be accessible for first-year graduate students. The explanations are a little more detailed than those of, say, Broecker & Jaenich's Introduction to Differential Topology, although not to the point of the spoon-feeding ones finds in Lee or Guillemin & Pollack, so students with no prior exposure to manifold topology may find that it moves a little too fast (all the material mentioned above fits into the first 165 pages). There's a 20-page refresher appendix on differential analysis, covering the prerequisites for the book, such as metric spaces, Banach spaces, (an unhelpful definition of) tensor products, the inverse function theorem, Sard's theorem, etc., although (1) much of it is too brief to help you if you don't know it already, (2) not all of it is really necessary for this book (e.g., don't worry if you don't know what a Banach space is), and (3) the full proof (modulo some mistakes) of Sard's theorem is given, so you don't need to have learned it elsewhere. There are between 2 and 8 exercises at the end of each chapter (relatively few when compared to other introductory texts), whose level ranges from the routine to the very difficult, but this is because they are "intended for students to work at and then discuss with a supervisor." However, "for the benefit of readers working independently," a chapter of solutions for the exercises is included, which is big plus.

The main problem with the book is that it tries to do too much, as there is no topic here that is not covered better in some other book, even though no other book covers all this material. Some of the chapters are way too brief, with the most egregious examples being that on Lie groups and Lie algebras (12 pages to cover both, including maximal tori and cohomologies of compact Lie groups), de Rham cohomology (it would really help if the reader has some exposure to algebraic topology first), and Morse theory and handle decompositions. One byproduct of squeezing in too many topics is that the proofs start to become rushed in the last couple of chapters, with lots of handwaving and some mistakes, as the care and precision in definitions and notation of the earlier chapters evaporates. Some of the biggest mistakes/weaknesses in this insufficiently copyedited book include: an overly restrictive proof of the Whitney embedding theorem; a mix-up in the proof of Stokes's theorem between equations on the manifold and those in the local coordinate neighborhood; at least 6 mathematical typos in the equations for Lemma 5.5.1; and 6 serious errors in the proof that every manifold admits Morse functions, including omitting the word "degenerate" in front of "critical point" and twice stating the exact opposite of what is intended(!). Furthermore, on p. 155 the authors state an important technical result on Lie subgroups and then prove only a special case to avoid dealing with the full technicalities, but implicit in their proof of the special case is actually an unrecognized use of the general case.

However, probably the worst mistakes occur on pp. 143-6 in an attempted proof of the Morse homology theorem that includes "open" sets that are not open, a Mayer-Vietoris cover that fails to cover the manifold in question (and a good thing, too, because if it did the formula would be wrong), and a diagram (Fig. 7.6) that clearly does not match the text. (You will have to learn Morse theory and handle decompositions from a different book, such as Hirsch's Differential Topology or Kosinski's Differential Manifolds, although even elementary books such as Gauld's Differential Topology: An Introduction or Wallace's Differential Topology: First Steps are better.) As a final example of a botched proof, I mention Sard's theorem, which the authors commendably attempted to state and prove for the minimal differentiability required. Unfortunately, what they ended up doing was stating the theorem properly in the case where the domain and range spaces have equal dimension, but then making a mistake in the estimation of the constants in the proof (cf. de Rham's proof, in Varietes differentiables, which they attempted to imitate, but missed), and misstating the theorem in the case where the dimensions are unequal, but then presenting the correct (except for a tiny mix-up) proof (from Milnor's Topology from the Differentiable Viewpoint) for the smooth case. See Sternberg's Lectures on Differential Geometry for a proper proof of the full theorem.

On the other hand, a few things are handled particularly well in the book: The various (5) definitions for tangent vectors (although I prefer Broecker & Jaenich's treatment, which these authors cite as an inspiration for this book), the proofs of the Poincare lemma and Mayer-Vietoris theorem in the chapter on cohomology, and degree theory (namely, degrees of maps, indices of vector fields, winding numbers, the Euler characteristic, and the Poincare-Hopf theorem that relates them; similar to treatments in Milnor and Guillemin & Pollack). The most startling feature of this book is the additional topics it covers; in particular, in Chapter 3 on fibre bundles, there's a section on "applications" that presents Thurston's Geometrization Conjecture (or should I say theorem now?) for 3-dimensional manifolds, along with a motivating analogy from the familiar 2-dimensional situation and a partial explanation of some of the terms involved (such as "pseudo-Anosov"), although most of this discussion will be way over the readers' heads, and there is understandably no mention of the method of proof. Even more surprising is what follows: A statement of Dennis Barden's own classification theorem for simply connected, closed, smooth 5-manifolds, which to my knowledge has not appeared outside his original papers. Unfortunately, the theorem is only then related to some Brieskorn varieties, about which is given not much more than a definition and an example. This material should really have been reserved for a more advanced text.

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

Product Description
In The Equilibrium Manifold, noted economic scholar and major contributor to the theory of general equilibrium Yves Balasko argues that, contrary to what many textbooks want readers to believe, the study of the general equilibrium model did not end with the existence and welfare theorems of the 1950s. These developments, which characterize the modern phase of the theory of general equilibrium, led to what Balasko calls the postmodern phase, marked by the reintroduction of differentiability assumptions and the application of the methods of differential topology to the study of the equilibrium equation. Balasko's rigorous study demonstrates the central role played by the equilibrium manifold in understanding the properties of the Arrow-Debreu model and its extensions. Balasko argues that the tools of differential topology articulated around the concept of equilibrium manifold offer powerful methods for studying economically important issues, from existence and uniqueness to business cycles and economic fluctuations.

After an examination of the theory of general equilibrium's evolution in the hundred years between Walras and Arrow-Debreu, Balasko discusses the properties of the equilibrium manifold and the natural projection. He highlights the important role of the set of no-trade equilibria, the structure of which is applied to the global structure of the equilibrium manifold. He also develops a geometric approach to the study of the equilibrium manifold. Applications include stability issues of adjustment dynamics for out-of-equilibrium prices, the introduction of price-dependent preferences, and aspects of time and uncertainty in extensions of the general equilibrium model that account for various forms of market frictions and imperfections. Special effort has been made at reducing the mathematical technicalities without compromising rigor.

The Equilibrium Manifold makes clear the ways in which the postmodern developments of the Arrow-Debreu model improve our understanding of modern market economies.

Arne Ryde Memorial Lecture Series ... Read more

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CR Manifolds and the Tangential Cauchy Riemann Complex provides an elementary introduction to CR manifolds and the tangential Cauchy-Riemann Complex and presents some of the most important recent developments in the field. The first half of the book covers the basic definitions and background material concerning CR manifolds, CR functions, the tangential Cauchy-Riemann Complex and the Levi form. The second half of the book is devoted to two significant areas of current research. The first area is the holomorphic extension of CR functions. Both the analytic disc approach and the Fourier transform approach to this problem are presented. The second area of research is the integral kernal approach to the solvability of the tangential Cauchy-Riemann Complex. CR Manifolds and the Tangential Cauchy Riemann Complex will interest students and researchers in the field of several complex variable and partial differential equations. ... Read more

Product Description
This unusual book, richly illustrated with 19 colour platesand about 250 line drawings, explores the relationshipbetween classical tessellations and3-manifolds. In hisoriginal entertaining style with numerous exercises andproblems, the author provides graduate students with asource ofgeomerical insight to low-dimensional topology,while researchers in this field will find here an account ofa theory that is on the one hand known tothem but here ispresented in a very different framework. ... Read more

Customer Reviews (1)

Solves the Alhambra mystery!
In 1944, Edith Muller claimed that all the 17 wallpaper groups were found in Alhambra. Unfortunately, her paper only contained 11 of them. Grunbaum and Shephard found two more, and finally in 1987 Perez-Gomez found the missing 4.

The book includes pictures of all the 17 wallpaper groups from Alhambra.

It should however be pointed out that in his review of the paper by Perez-Gomez in Mathematical Reviews, Coxeter disputes the p3m1 example. ... Read more

Product Description
This book, together with its companion volume Design Techniques for Engine Manifolds – Wave Action Methods for IC Engines, reports the significant developments that have occurred over the last twenty years and shows how mature the calculation of one-dimensional flow has become. In particular, they show how the application of finite volume techniques results in more accurate simulations than the ‘traditional’ Method of Characteristics and gives the further benefit of more rapid and more robust calculations.

CONTENTS INCLUDE:

• Introduction
• Governing equations
• Numerical methods
• Future developments in modelling unsteady flows in engine manifolds
• Simple boundaries at pipe ends
• Intra-pipe boundary conditions
• Turbocharging components
• The application of wave action methods to design and analysis of flow in engines.

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Four papers by C.H. Taubes (Harvard University) comprising the complete proof of his remarkable result relating the Seiberg-Witten and Gromov invariants of symplectic four manifolds.2010 paperback re-issue. ... Read more

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This book presents the classical theorems about simply connected smooth 4-manifolds: intersection forms and homotopy type, oriented and spin bordism, the index theorem, Wall's diffeomorphisms and h-cobordism, and Rohlin's theorem. Most of the proofs are new or are returbishings of post proofs; all are geometric and make us of handlebody theory. There is a new proof of Rohlin's theorem using spin structures. There is an introduction to Casson handles and Freedman's work including a chapter of unpublished proofs on exotic R4's. The reader needs an understanding of smooth manifolds and characteristic classes in low dimensions. The book should be useful to beginning researchers in 4-manifolds. ... Read more

Customer Reviews (1)

Excellent
For those genuinely interested in understanding the proof of the 4-dimensional Poincare conjecture, and for those who need a more geometric, intuitive view of some of the main results in topological 4-manifolds, rather than one based on the heavy machinery of algebraic topology, this book is an excellent beginning. The author endeavors in this book to be as clear as possible, and he does not hesitate to use diagrams to get the point across. Rigor however, is not sacrificed. One of the main goals in the book is to get a more geometric proof of Rohlin's theorem, which states that cobordism ring in 4-dimensions over the special orthogonal group and over the spin group is the integers.

The author starts the book with an overview of handlebody theory, noting that for the case of interest, 4-dimensional toplogical manifolds must be smooth in order for them to be handlebodies. Smooth handlebody decompositions can be described by Morse theory, and one smooth handlebody decomposition can be related to another via an isotopy of attaching maps and creation or annihilation of handle pairs. The author visualizes handlebodies in four dimensions by drawing their attaching maps in the 3-sphere. This results in the use of framed links to model the attaching maps, with examples of the 3-torus, the Poincare homology 3-sphere, and a homotopy 4-sphere, the latter of which is homeomorphic to the 4-sphere and is a double cover of an exotic smooth structure on 4-d real projective space. The author also gives a brief but interesting discussion on why the methods of this chapter are difficult to do in three dimensions.

The theory of intersection forms appears in chapter two, with the author proving first that for a closed, smooth, oriented, 4-d manifold M any element of the second integer homology group is represented by a smoothly imbedded oriented surface. Any two such surfaces can be joined by smooth oriented 3-manifold imbedded in M. The isomorphism between the second homology and cohomology groups (over the integers) modulo torsion is the famous "intersection pairing". The author then proves that two simply-connected, closed, oriented 4-manifolds are homotopy equivalent if and only if their intersection forms are isometric. The proof emphasizes the geometric connection between homotopy type and intersection forms. A brief review of symmetric bilinear forms and characteristic classes is then given, as preparation for the classification results given later in the book.

The author treats classification theorems in chapter three, which he describes as deciding which forms, whether symmetric, integral, or unimodular, can be represented by simply connected closed 4-manifolds. The relation between forms and homotopy type makes this implicitly a classification for the homotopy type of the manifold. Rohlin's theorem was historically the first major result in this problem, but the author delays its proof until chapter eleven. The author briefly discusses the work of Freedman in the topological case, and Donaldson, in the smooth case.

Spin structures are discussed in chapter four and several examples are given. The author also shows how to relate spin structures on the boundary of a manifold to spin structures on the manifold itself, to set up later discussions on cobordism. Chapter five then concentrates on the Lie group spin structure of the 3-torus T3(Lie) and the surface constructed by taking the nine-fold direct sum of complex 2-d projective space and its reverse orientation. The latter is a complex analytic projection, which is a smooth fiber bundle with fiber the two-torus except for a finite number of singular fibers. The author shows in detail how to use this object to obtain a spin manifold with spin boundary T3(Lie).

Chapter six is devoted to showing how to immerse closed, smooth, oriented 4-manifolds in Euclidean 6-d space. This involves the calculation of a characteristic class in the second integral cohomology group. Then as a warm-up to showing that a spin 4-manifold with index zero spin bounds a spin 5-manifold, the author proves in chapter 7 that every orientable 3-manifold is spin, bounds an orientable 4-manifold, and if spin bounds a spin 4-manifold with only 0-handles.

In chapter eight, the author proves that a closed, smooth, connected, and orientable 4-manifold is the boundary of a smooth 5-manifold if the first Pontryagin class is 0. If the 4-manifold is spin, and the first Pontryagin class is 0, then there exists a smooth, spin 5-manifold whose boundary is the 4-manifold, where both manifolds are considered as spin manifolds. Chapter nine proves the Hirzebruch index theorem in dimension 4, and the author shows that the cobordism ring for SO and Spin is the integers. Chapter ten is devoted to a proof of Wall's theorem and the h-cobordism theorem in dimension 4. The geometric proof of Rohlin's theorem promised by the author is finally done in chapter eleven.

Casson handles, so important in the proof of the 4-d Poincare conjecture, are discussed in chapter twelve. The author shows the role of the Whitney trick in dimensions 5 or more, and how its failure in dimension 4 results in the use of Casson handles, which are constructed using the famous "finger moves". He gives an explicit handlebody description of the simplest Casson handle, and then relates it to the Whitehead continuum.

The most fascinating part of the book is chapter thirteen, which outlines briefly Freedman's proof of the 4-dimensional Poincare conjecture. The proof makes use of 4-dimensional handlebody theory and decomposition space theory. Casson handles are decomposed via an imbedding of a Cantor set of Casson handles inside them. The "Big Reimbedding theorem" of Freedman, which points to the existence of an exotic smooth structure on the 3-sphere cross the real line, is quoted but not proved. The book ends with chapter fourteen being a brief discussion of exotic structures, their existence following from the non-smoothness of Casson handles. ... Read more

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This account of basic manifold theory and global analysis, based on senior undergraduate and postgraduate courses at Glasgow for students and researchers in theoretical physics, has been proven over many years. The treatment is rigorous yet less condensed than in books written primarily for pure mathematicians. Prerequisites include knowledge of basic linear algebra and topology, the latter of which is included in two appendices, as many courses on mathematics for physics students do not include this subject. Topics covered include vector spaces; tensor algebra; differentiable manifolds; exterior differential forms; pseudo-Riemannian and Riemannian manifolds; sympletic manifolds; and complex linear algebra. ... Read more

Customer Reviews (1)

required reading
This is on John Sidles' list of "yellow" books and is required reading.You don't need to read all of it, he says, but you should be at the level where you're not afraid to open it and find something useful if you are doing pull-back and push-forward geometric quantum simulations. ... Read more

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These volumes are based on lecture courses and seminars given at the LMS Durham Symposium on the geometry of low-dimensional manifolds.This area has been one of intense research recently, with major breakthroughs that have illuminated the way a number of different subjects (topology, differential and algebraic geometry and mathematical physics) interact. ... Read more

Product Description
These volumes are based on lecture courses and seminars given at the LMS Durham Symposium on the geometry of low-dimensional manifolds.This area has been one of intense research recently, with major breakthroughs that have illuminated the way a number of different subjects (topology, differential and algebraic geometry and mathematical physics) interact. ... Read more

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Infinite dimensional manifolds, Lie groups and algebras arise naturally in many areas of mathematics and physics. Having been used mainly as a tool for the study of finite dimensional objects, the emphasis has changed and they are now frequently studied for their own independent interest.On the one hand this is a collection of closely related articles on infinite dimensional Kähler manifolds and associated group actions which grew out of a DMV-Seminar on the same subject. On the other hand it covers significantly more ground than was possible during the seminar in Oberwolfach and is in a certain sense intended as a systematic approach which ranges from the foundations of the subject to recent developments. It should be accessible to doctoral students and as well researchers coming from a wide range of areas.The initial chapters are devoted to a rather selfcontained introduction to group actions on complex and symplectic manifolds and to Borel-Weil theory in finite dimensions. These are followed by a treatment of the basics of infinite dimensional Lie groups, their actions and their representations. Finally, a number of more specialized and advanced topics are discussed, e.g., Borel-Weil theory for loop groups, aspects of the Virasoro algebra, (gauge) group actions and determinant bundles, and second quantization and the geometry of the infinite dimensional Grassmann manifold. ... Read more

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Customer Reviews (2)

Saved me in grad school
When I found this book in graduate school, I cried out O frabjous day! Callooh! Callay!There were a number of ideas that I just couldn't seem to grasp.Everyone else kept using these words, and they just kept slipping out of my head every time I thought I understood them.Then I found this little gem and I felt like someone had turned on the lights.I was astonished to see that it went out of print.When I saw that Dover reprinted it, I bought a new copy.After reading the other review, I feel like a dinosaur.But I did go to graduate school a long long time ago...

out-of-date; even worse, sometimes incorrect
Auslander & MacKenzie's "Introduction to Differential Manifolds" was one of the first books on differential manifolds (the back cover actually claims it to be The first, but I believe Munkres' Elementary Differential Topology was earlier, and certainly Milnor's and Hu's published lectures notes were), so perhaps it is no surprise that it is out-of-date. But the terminology and definitions differ so significantly from modern ones, this book can actually harm a new graduate student by implanting false concepts that will then have to be unlearned.

There is a nice selection of topics: definitions of manifolds, diffeomorphisms, submanifolds, etc.; submanifolds of Euclidean space and projective varieties; Lie groups and algebras; principal and fibre bundles; multilinear (i.e., tensor) algebra; the Whitney embedding theorem; and foliations and the Frobenius theorem. Some of these are well presented, at an easy level for beginners - the sections on projective varieties, fibre bundles, and foliations in particular are not usually found (or, at least, this well explained) in introductory differential topology textbooks, so it would almost be worth reading this book for these chapters. And there are slightly unusual treatments of some things, such as the implicit function theorem and the differential of a function (defined as an equivalence class of functions).

However, most subjects are treated rather cursorily, often devoting the bulk of the chapter to bland formalism or basic definitions and not getting to anything actually interesting. A good example of this is the last chapter, on tensor products: 25 pages are used to develop the formal theory of tensor and exterior products, as one would encounter in an algebra book such as Lang, and then only on the last page is the exterior derivative introduced, with no time to then use differential forms for anything (e.g., no mention of integration). Even worse, the first pages of the book begins with a discussion of R^n that separates out the vector space and metrical properties, using different letters to denote R^n as a metric space (E^n) and as a vector space (V^n), and then continues this needless distinction throughout the book. The notion of "attaching" V^n to R^n is not found anywhere else that I've seen and seems to be one of those pedagogical approaches that never caught on and serves no purpose for modern readers.

Even the sections that are well written are very inadequate. The chapter on foliations doesn't actually use the term (or the definition of) foliations anywhere, nor does it tell you that it has in fact proved the Frobenius theorem. The chapter on Lie groups (and algebras) doesn't even mention any of the classical ones except the orthogonal groups and GL(n). The explanations of projective space and projective varieties are nice, but only scratch the surface of algebraic geometry. Vector bundles and Riemann metrics are defined and a few properties are demonstrated, but they are not used again elsewhere. There are brief treatments of flows and partitions of unity, but the latter in particular are hardly used when compared with most diff top books. There are relatively few graphs (about a dozen in the whole book) and the exercises are generally pretty easy - many of them are just filling in missing steps in the proofs.

But the worst feature of this book is the fact that some of the definitions of important concepts such as diffeomorphism and submanifold are different from current usage. In fact, they differ from even earlier works, such as Milnor's Differential Topology notes (in Collected Papers of John Milnor. Volume III: Differential Topology), as they are using the older definition for a submanifold from differential geometry, but mix in results from Milnor, without even realizing the discrepancies (cf. Munkres for a discussion of this issue). This is evidenced by the fact that diffeomorphism is actually defined 2 different ways, with the authors seemingly unaware that an injective immersion is not necessarily an embedding. Their "submanifold" also has the property that it may not share the same topology from the manifold of which it is a submanifold! In fact, on pp. 89-90 they construct a "submanifold" of the torus, via an injective immersion of the line (and a 2-page proof that rationals are dense in the reals), that has a different topology, whereas Kosinski (Differential Manifolds) on p. 27 uses the exact same example to show that this subset of the torus is NOT a submanifold! (To be fair, Bishop & Crittenden's "Geometry of Manifolds," from the same time period, also makes this "mistake," but they were at least writing a diff geom book.) A similar thing happens in the proof of the Whitney "embedding" theorem, where Milnor's proof is followed almost to the letter, with a few more details filled in, but one exercise that Milnor leaves for the reader has been omitted, so that instead of proving an embedding theorem, the authors only succeed in proving the existence of an injective immersion, which they mistakenly call a diffeomorphism. Then there are other differences in terminology, such as calling an immersion a regular function or defining a Lie group to be real-analytic (and then having to demonstrate this property explicitly).

There are only a few typos, but the print quality is so poor (it looks like it was made from a photograph of the old pages) that sometimes one mistakes one number or letter for another, since the text is so washed out.

In short, if you want to learn about differential manifolds, give this book a pass and buy instead Broecker & Jaenich's Introduction to Differential Topology, Lee's Introduction to Smooth Manifolds, Gauld's Differential Topology: An Introduction, Guillemin & Pollack's Differential Topology, Barden & Thomas's An Introduction to Differential Manifolds, Hirsh's Differential Topology, or Milnor's Topology from the Differentiable Viewpoint. None of these books has quite the same emphasis, so you'll still need to learn Lie groups & algebras and projective varieties elsewhere, but at least you'll learn everything right the first time.

... Read more

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Einstein's equations stem from General Relativity. In the context of Riemannian manifolds, an independent mathematical theory has developed around them. Recently, it has produced several striking results, which have been of great interest also to physicists. This Ergebnisse volume is the first book which presents an up-to-date overview of the state of the art in this field. "Einstein Manifold"s is a successful attempt to organize the abundant literature, with emphasis on examples. Parts of it can be used separately as introduction to modern Riemannian geometry through topics like homogeneous spaces, submersions, or Riemannian functionals.

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The book starts with a thorough introduction to connections and holonomy groups, and to Riemannian, complex and Kähler geometry.Then the Calabi conjecture is proved and used to deduce the existence of compact manifolds with holonomy SU(m) (Calabi-Yau manifolds) and Sp(m) (hyperkähler manifolds).These are constructed and studied using complex algebraic geometry.The second half of the book is devoted to constructions of compact 7- and 8-manifolds with the exceptional holonomy groups 92 and Spin(7). Many new examples are given, and their Betti numbers calculated.The first known examples of these manifolds were discovered by the author in 1993-5.This is the first book to be written about them, and contains much previously unpublished material which significantly improves the original constructions. ... Read more

Customer Reviews (1)

WELL DONE!
This book is, as the author mentioned in the preface, a marriage of two parts. The first part provided more or less a self-contained introduction to the theory of Riemannian holonomy groups, which usually couldn't be found in differential geometry textbooks. The second part is a research monograph on exceptional holonomy groups, which is the subject that the author is famous at. This book contains lots of topics which are hard to be found in any other books. For example, it contains a proof of the Calabi conjecture, which I've never seen in anywhere else except Yau's original papers. It also has a concise introduction to Calabi-Yau manifolds, which includes lots of topics about CY manifolds that are hard to be found in just a single book. Overall, it's a great introduction to the theory of holonomy groups. And also provides a good start about differential geometric side of the theory of Calabi-Yau manifolds, together with a roughly complete list of further references. ... Read more

Product Description
Our first knowledge of differential geometry usually comes from the study of the curves and surfaces in $I\!\!R^3$ that arise in calculus. Here we learn about line and surface integrals, divergence and curl, and the various forms of Stokes' Theorem. If we are fortunate, we may encounter curvature and such things as the Serret-Frenet formulas. With just the basic tools from multivariable calculus, plus a little knowledge of linear algebra, it is possible to begin a much richer and rewarding study of differential geometry, which is what is presented in this book. It starts with an introduction to the classical differential geometry of curves and surfaces in Euclidean space, then leads to an introduction to the Riemannian geometry of more general manifolds, including a look at Einstein spaces. An important bridge from the low-dimensional theory to the general case is provided by a chapter on the intrinsic geometry of surfaces. The first half of the book, covering the geometry of curves and surfaces, would be suitable for a one-semester undergraduate course. The local and global theories of curves and surfaces are presented, including detailed discussions of surfaces of rotation, ruled surfaces, and minimal surfaces. The second half of the book, which could be used for a more advanced course, begins with an introduction to differentiable manifolds, Riemannian structures, and the curvature tensor. Two special topics are treated in detail: spaces of constant curvature and Einstein spaces. The main goal of the book is to get started in a fairly elementary way, then to guide the reader toward more sophisticated concepts and more advanced topics. There are many examples and exercises to help along the way. Numerous figures help the reader visualize key concepts and examples, especially in lower dimensions. For the second edition, a number of errors were corrected and some text and a number of figures have been added. ... Read more

Customer Reviews (5)

elegant work
The author gives a clean and wise introduction to the three major parts in differential geometry-curves-surfaces-manifolds. The important concepts in classic results were introduced by short but fully content paragraphs.

The author wrote no gossip in the context and always touch the ideas with a niddle; therefore I should follow that:

This is the best book for introducing differential geometry.

Fast moving
This is a very fast moving book, covering a huge amount of material at a fairly sophisticated level in under 380 pages. For example, differential forms are introduced in about 2 pages so that the Maurer-Cartan structural equations can be defined.The first 4 chapters makes up a very concise course in curves and surfaces, while the last 4 chapters cover Riemannian geometry.In comparison, do Carmo's two books take 500 pages for the former and 320 pages for the latter.

For this reason I think the claim that this could be used as an undergraduate text is overly optimistic.For that I would use a more self-contained text like Millman & Parker (ISBN: 0132641437).But it would make an excellent text for a graduate survey, or as a second text for someone wanting to make the transition from classical theory (learned from, say, one of the Dover books like Struik, ISBN: 0486656098) to more modern methods.Also, you'll probably want to supplement with a gentler introduction to differential forms.

Of interest to students of physics, the book covers curves and surfaces in Minkowski space, as well as Einstein spaces.

A excellent introduction for the 21st century
While there is exist many classic texts on differential geometry, I have particularly appreciated this book for its up-to-date treatment, numerous well-done figures, broad coverage, elegant type-setting, and clear expositions. The book covers all the basics expected from an introduction to differential geometry, including curves and 2-D surfaces, but with a look towards the more advanced material in the second half of the book. It alternates between Ricci style notation and Koszul style notation, often carefully explaining the relation between the two and giving examples (I found this particularly helpful). There are, however, some sections where the english is a bit rough (perhaps the fault of the translator). It is also quite brisk throughout, often mentioning advanced topics before they are treated in detail. For example, it already mentions submanifolds, tangent spaces, and tangent bundles in the first chapter on "Notations and Prerequisites from Analysis." It will require serious attention, especially if one has not encountered a good dose of abstract mathematics before. Nonetheless, I have found myself returning to it over several years as an excellent reference and source of many additional topics that I skipped on a first reading. For example, the final chapter on Einstein spaces is a valuable, though demanding, bonus. Thanks to the AMS for publishing a fine edition of a top-notch German author's work.

A beautiful geometry
This book is very useful for students who are interested in geometry. The book is organized from elementary facts to advanced geometry very well. This book provides to students thereason why they study the geometry. This book explains very easily that the geometry of curves and surfaces can be generalized to high dimensional Riemannian manifolds naturally.
Moreover, the edition of this book is very beautiful and helpful for readers. For example, the important results are placed in boxes.

Attractive book on differential geometry
Differential geomety is perhaps the most beautiful part of higher mathematics. It combines geometry, analysis and intuition in a wonderful way. This attractive book is a concise and modern book that manages to be both pedagogical and accurate in a pleasant way. In only 350 pages most of the differential geometry that a non-expert will ever need is outlined. Illustrations and notation seem optimal for their purpose. The book is a worthy successor of classics like Struik, Stoker, and Kreyzig. ... Read more