Product Description
Theodore Frankel explains those parts of exterior differential forms, differential geometry, algebraic and differential topology, Lie groups, vector bundles and Chern forms essential to a better understanding of classical and modern physics and engineering. Key highlights of his new edition are the inclusion of three new appendices that cover symmetries, quarks, and meson masses; representations and hyperelastic bodies; and orbits and Morse-Bott Theory in compact lie groups. Geometric intuition is developed through a rather extensive introduction to the study of surfaces in ordinary space.First Edition Hb (1997): 0-521-38334-XFirst Edition Pb (1999): 0-521-38753-1 ... Read more

Customer Reviews (18)

could have been so much better
this is a damn hard book to understand. i am a theoretical physicist who already knows the stuff here and i find the explanations sometimes very confusing.it's not a simple book to read. you have to pay close attention to every line, and this is a major pain since the book is so big. also some sentences are very bloated. i know the author spent a lot of time on this book. but he should have gotten a better copy editor.

it has a lot going for it.the emphasis on physics is nice.but unless you have a lot of patience and haven't read any other books (so you don't confused about the unorthodox notation), this book may drive you crazy.

its really too bad, because this is the only book that i know of which tries to really "explain" bundles in a non-pedantic way.unfortunately, it is not that successful. (the part on bundles is perhaps the most transparent and easiest part of the book though).

if you have patience and are willing to decipher some of the prose which is ambiguous, or difficult to understand because there are like 5 commas in a single sentence, then this is absolutely a 5 STAR book.the material is wonderful, and there is an enormous amount of insight shared.

otherwise, if you are a typical math guy (like me) you are probably better off going with a thinner and more mathematical book like Darling's differential geometry book or Morita's book or the new book on differential geometry by Tu.

it is of course an alternative to Nakahara.but Nakahara isn't a proper textbook.it's a collection of examples. in summary, despite the book's flaws, i have not seen anyone write a better story in a single volume of how to combine math and physics at an "elementary" level than Frankel.

Fantastic - for the scientist
A very good book: buy it. But only if you are a scientist or student of physics/mathematics. This is not popular-science-common-public level.

a book worth keeping
This book can be quite confusing if you start without any background on the idea of manifold or knows nothing about general relativity. However, it does have strong points:

1. The notation is very up-to-date, and is entirely coordinate-independant approach.

2. The author explains in great details of formulation of modern differential geometry, and the details are comparatively lacking in other reference books.

3. The author never hesitate to use graphs and diagrams to illustrate points, and stroke nice balance in between mathematics rigor and physical insight.

Although it appears quite verbose at some point, it is mainly because differential geometry is such a heavy subject. Another book nice to have as companion reading is Goldburg's "Tensor analysis on Manifold", a terse, well-written text book.

Phenomenal
I just finished reading this book and I found it phenomenal. The physical ideas are made very clear in a natural mathematical framework.

You should buy this, despite its flaws
The other reviews on this page give this book anywhere from 1 to 5 stars, and they are all correct in their own way. The book is inspired, deep and full of physics applications and insights. On the other hand, it skims over mathematical rigor to a large degree and focuses more on defining things, getting a feel for them and moving on to application.

My advice: buy the book for its strengths, and read other books in parallel if you need more rigor. But still, buy it.

Also, things can be confusing on the first two or three reads, but keep at it and you will be glad you did. ... Read more

Product Description
Differential geometry has a long, wonderful history. It has found relevance in areas ranging from machinery design to the classification of four-manifolds to the creation of theories of nature's fundamental forces to the study of DNA. This book studies the differential geometry of surfaces with the goal of helping students make the transition from the compartmentalized courses in a standard university curriculum to a type of mathematics that is a unified whole. It mixes together geometry, calculus, linear algebra, differential equations, complex variables, the calculus of variations, and notions from the sciences. Differential geometry is not just for mathematics majors. It is also for students in engineering and the sciences. The mix of ideas offer students the opportunity to visualize concepts through the use of computer algebra systems such as Maple. The book emphasizes that this visualization goes hand-in-hand with the understanding of the mathematics behind the computer construction. Students will not only see geodesics on surfaces, but they will also observe the effect that an abstract result such as the Clairaut relation can have on geodesics. Furthermore, the book shows how the equations of motion of particles constrained to surfaces are actually types of geodesics. The book is rich in results and exercises that form a continuous spectrum, from those that depend on calculation to proofs that are quite abstract. ... Read more

Customer Reviews (3)

clearest undergrad differential geometry text around
This is a very well-written text on modern differential geometry for undergraduates. The content of the book is similar to O'Neill's "Elementary Differential Geometry" (e.g. covariant derivatives, shape operators), but it's easier to read. There are many undergrad texts around -- O'Neill, do Carmo, Pressley -- but this one is the most lucidly written one hands-down.

Afer going through Oprea, one might like to tackle O'Neill's "Elementary Differential Geometry" and Vols 2-4 of Spivak's "Comprehensive Introduction to D.G."

Like O'Neill, Oprea develops surface theory using the shape operator. But Oprea takes shortcuts and doesn't develop the theory in quite the same generality as O'Neill does. For example, Oprea doesn't introduce differential forms and the exterior calculus. As a consequence, Oprea restricts himself to the Serret-Frenet equations whereas O'Neill introduces Cartan's structural equations -- of which Serret-Frenet is simply a special case -- as the method of moving frames in full generality. The structural equations are then used (by O'Neill) in both curve and surface theory.

Nice introduction and applications of differential geometry
I found this book to be a fine introduction to this subject. I was particularly pleased with the practical examples outlined in the book. Even though I am not extremely proficient with Maple, I found the exercises using this software provided important illustrations of applications.

Not a text for a rigorous mathematics course
This book is not to be used as a rigorous introduction to differential geometry.There are some definitions and theorems that are casuallydescribed, and the motive behind particular definitions are vague. Thosenot interested in MAPLE might find constant instructions for MAPLEannoying. Not to be completely negative, there are some good excercizes inthe text that I especially enjoyed. ... Read more

Product 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

Customer Reviews (5)

An excellent transition for the beginning grad student
I agree that this work is bit to terse for those completely uninitiated with the subject. However, for a first semester graduate student (or anyone with about this level of maturity) who has at least a minimal acquaintance with curves/surfaces this is a wonderful book which fulfills a long standing gap. The divide between undergraduate differential geometry and graduate geometry is too great. A beginning graduate student walk into a course on differentiable manifolds and Riemannian Geometry with no undergraduate diff geometry and be fine (as I did). There simply doesn't seem to be anymore that a superficial connection (no pun intended) between the two. This may not trouble the student at first, but as Spivak notes in his tome: "this ignorance of the roots of the subject has its price." Eventually one needs to assimilate the intuition of classical geometry with the technical language of manifolds. This book very elegantly leads the student from his undergraduate education to the doorstep of modern global geometry. As an added bonus, the author also endeavors to bridge the gap between Euclid and differential geometry.

The only other successful attempt at this is in Spivak, but unfortunatelyhe goes backwards. Volume 1 is entirely devoted to manifolds. Then in volume 2 he explains the classical point of view and then builds the bridge. While these are beautiful books, this is not efficient for the beginning student. The prospective geometer should read this before a class on manifold theory.

I give 4 stars only because the author advertises this as a "first exposure", and this book is simply not suited for this purpose.

great history of geometry book, terrible introductory differential geometry book
Do not buy this inappropriately titled book if you are seeking an introductory text to learn differential geometry. It's not that the concepts in the book are so advanced, so much as not that much space is actually devoted to the subject. The author's real objective is to trace the development of geometry from Euclid to the (relatively) modern formulation of differential geometry, and as a book on that topic it succeeds admirably.

The core theme of the book is that efforts to prove the parallel postulate, or, equivalently, show that non-Euclidean geometries are impossible, inadvertently, through their failure, led to the discovery of many fascinating areas of mathematics, such as hyperbolic and Riemannian geometries, and to the development of philosophical ideas about what actually constitutes mathematics and how it is independent from physical reality. The book culminates with the results of Beltrami and Poincare that showed that hyperbolic and Euclidean geometries are logically equivalent, in the sense that if there is a self-contradiction in one then the other is also impossible, thus putting an end to all attempts to disprove hyperbolic geometry. (Unfortunately, Marilyn vos Savant is unaware of this, or at least she was when she wrote an article some years back criticising Andrew Wiles's proof of Fermat's last theorem because it used hyperbolic geometry.)

As an appendix, McCleary adds a translation of Riemann's lecture "On the hypothesis which lie at the foundations of geometry," perhaps the most influential single lecture in the history of mathematics (and physics), in which, in the mid-1860s, he presented to a general faculty a talk (involving only a single equation) on the foundations of geometry that anticipated the concepts of a manifold and Riemannian geometry as well as general relativity and even hinted at quantum mechanics.

I used this text as a primary reference when conducting an undergraduate seminar on the history of hyperbolic geometry 12 years ago. For this purpose it was suited perfectly, but if you want to learn differential geometry by all means buy one of do Carmo's books or Gallot, Hulin, and LaFontaine.

not for the uninitiated
I'm a master's student in math. I bought the book thinking I'd use it for an independent study. I was wrong.

The book has interesting historical tidbits and some classical proofs, including material I hadn't seen elsewhere. However, it takes little time to explain to the novice exactly what's going on. It comes off more as a set of lecture notes than as a text for self-study.

For instance, in ch. 8 McCleary breezes through the basics of regular surfaces--coordinate charts, differentiability, implicit/inverse function theorem, the tangent space, orientability, the first fundamental form in about 19 pages. This is the same foundational material that folk like do Carmo or O'Neill rightfully spend 60-70 pages to cover.

His treatment of the Gauss map and the second fundamental form is even more schematic.

If I hadn't already worked the other books, when I got to McCleary's treatment of surfaces I would've been completely lost.

This book is best for people who know basic differential geometry already but are curious about certain historical aspects of it, not for people who are trying to learn differential geometry.

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

Product Description
Ideas of projective geometry keep reappearing in seemingly unrelated fields of mathematics. This book provides a rapid route for graduate students and researchers to contemplate the frontiers of contemporary research in this classic subject. The authors include exercises and historical and cultural comments relating the basic ideas to a broader context. ... Read more

Product Description
The uniqueness of this text in combining geometric topology and differential geometry lies in its unifying thread: the notion of a surface. With numerous illustrations, exercises and examples, the student comes to understand the relationship between modern axiomatic approach and geometric intuition. The text is kept at a concrete level, 'motivational' in nature, avoiding abstractions. A number of intuitively appealing definitions and theorems concerning surfaces in the topological, polyhedral, and smooth cases are presented from the geometric view, and point set topology is restricted to subsets of Euclidean spaces. The treatment of differential geometry is classical, dealing with surfaces in R3 . The material here is accessible to math majors at the junior/senior level. ... Read more

Customer Reviews (5)

Great Book
It is a very intiutive book in both areas. Also at the end of the book there is a good material for further study, author explains the research fields in Geometry/Topology and related books. If you are an undergraduate and want to get an overall idea about the gradute study in topology and geometry that is a nice introduction.

a remark on omissions
I have not read the book, only the reviews.In one excellent review here it is remarked that it is "unfortunate" that the author does not prove the Schoenflies theorem and the triangulability of surfaces.

later this same reviewer observes that the proof of the smooth Gauss Bonnet theorem in the book seems relatively hard.I merely wish to point out that the author has made choices in the reader's interest both by what he includes and what he omits.

The two theorems named above which are not proved, could well take another entire book to prove.They are far harder than the smooth Gauss Bonnet theorem.

I have seen entire books devoted to proving triangulability, and Schoenflies theorem was the subject of weeks of tedious work in a topology course I took as a student.I still dislike even hearing of this result.So if these omissions are the reviewer's only criticisms of the book, they should rightly be considered pluses.

Hence I also give the book at least 4 stars, by logical deduction.

Good introduction
This book is suitable for reading at an advanced undergraduate or beginning graduate level. The author is careful to present the subject from both a rigorous point of view and one that emphasizes the geometric intuition behind the subject. These two approaches to teaching topology are not mutually exclusive, with this book giving a good example of this.

After a brief overview of the elementary topology of subsets of Euclidean space in chapter 1, topological surfaces are discussed in chapter 2. Surfaces are built up from arcs, disks, and one-spheres. Unfortunately, the proofs of the theorem of invariance of domain and the Schonflies Theorem are not included, but references are given. Gluing techniques though are effectively discussed, and the author does not hesitate to use diagrams to explain the relevant concepts. The more popular constructions in surface topology, namely the Mobius strip and the Klein bottle are given as examples of the cutting and pasting techniques. The amusing fact that the Klein bottle can be obtained from gluing two Mobius strips along their boundaries is proven.

The theory of simplicial surfaces is discussed in the next chapter. Simplicial surfaces are much easier to deal with for beginning students of topology. Simplicial complexes are introduced first, and the author then studies which simplicial complexes have underlying spaces that are topological surfaces. He proves that this is the case when each one-dimensional simplex of the complex is the face of precisely two two-dimensional simplices, and the underlying space of each link of each zero-dimensional simplex of the complex is a one-dimensional sphere. Unfortunately, the author does not prove that any compact topological surface in n-dimensional Euclidean space can be triangulated. The Euler characteristic is defined first for 2-complexes and it is shown that it is the same for two simplicial surfaces that triangulate a compact topological surface. The author does prove in detail the classification of compact connected surfaces. Interestingly, the author also proves a simplicial analogue of the Gauss-Bonnet theorem, and gives a proof of the Brouwer fixed point theorem.

The author turns to smooth surfaces in the next few chapters, wherein curves are defined along with the relevant differential-geometric notions such as curvature and torsion. The fundamental theorem of curves is proven. The reader is first introduced to the concept of what in more advanced treatments is called a differentiable manifold, and several concrete examples are given of smooth surfaces. The differential geometry of smooth surfaces is outlined, with the first fundamental form and directional derivatives discussed in great detail. The reader should be familiar with the inverse function theorem to appreciate the discussion of regular values.

Even more interesting differential geometry is discussed in chapter 6, which covers the curvature of smooth surfaces. The important Gauss map is defined, along with the Weingarten map and the second fundamental form. This allows an intrinsic notion of curvature, but the author does perform explicit computations of curvature using various choices of coordinates. The proof that Gaussian curvature is intrinsic (Theorema Egregium) is proven, along with the fundamental theorem of surfaces. Geodesics, so important in physical applications, are discussed in the next chapter. The reader gets a first look at the "Christoffel symbols", even though they are not designated as such in the book.

The book ends with a thorough treatment of the Gauss-Bonnet theorem for smooth surfaces. The smooth case is much more difficult to prove than the simplicial case, as the reader will find out when studying this chapter. The author also gives a very brief introduction to non-Euclidean geometry.

Presentation of The Spirit...
Lots of times the mathematicians stuck in proofs,in that fool symbols, forgetting the ideas, the picture.One can never find the right way withclosed eyes.This book teaches to think, getting beyond the symbols. Ithas also useful advises about the research areas. The author made his Phdat Cornell with D.Henderson.A beautiful undergraduate text.

Just a mediocre book of lesser extent
First, the title reads fine. But there's a catch. This kind of title sounds like it covers all. The truth is. It ain't true. Second. The author's attitude. I'd rather say the author is talking to himself. ... Read more

Product Description
The second edition of this text has sold over 6,000 copies since publication in 1986 and this revision will make it even more useful. This is the only book available that is approachable by "beginners" in this subject. It has become an essential introduction to the subject for mathematics students, engineers, physicists, and economists who need to learn how to apply these vital methods. It is also the only book that thoroughly reviews certain areas of advanced calculus that are necessary to understand the subject.



Line and surface integrals
Divergence and curl of vector fields ... Read more

Customer Reviews (5)

Great book
Great introductory differential geometry text!I used this book to help me pass my qualifying exam.Yay Boothby!

This is a book for REAL mathematicians
This book is an wonderful introduction to Differential Geometry for the serious student of mathematics. However, it is not aimed at engineers, physicists or even applied mathaticians.
The author assumes the reader has an extensive knowledge of abstract algebra and at least one course in analysis. Likewise, he has chosen to emphasis applications of the subject to Lie Groups, homotopy theory, and group actions, rather than the physical applications that applied mathematicians are looking for. But, for the student of pure mathematics, this text is a great starting point into the rich world of differential geometry.
Also, while this book is an introduction and requires no previous knowledge of the subject, it covers enough ground to be followed up by such topics as the Gauss-Bonnet Theorem, the Cartan-Hadamard Theorem, Bonnet's Theorem, or Morse Theory.

When accountants and soldiers take interest in geometry.....
One day, accountants and soldiers may take an interest in differential geometry. If and when such a day comes to pass, this book will have a role to play. Until then, engineers, physicists and mathematicians alike have better alternatives, such as the inspiring texts, with complementary qualities, by Burke, "Applied Differential Geometry"; by do Carmo, "Riemannian Geometry", or by Spivak, "A Comprehensive Introduction to Differential Geometry".

Even more advanced books such as Lang's or Petersen's are more readable: in them the extra formalism brings the reward of more powerful results. Here the retentive attention to the trees at the expense of the forest is merely a barrier to entry for the uninitiated. This text's popularity in some areas of engineering must have played a role in the slow acceptance of Riemannian geometric methods.

Manuel Tenide

great introductory text
My first course on manifolds was based on this book,and I believe that it is the best introduction to the subject (especially for beginners). I thoroughly enjoyed it! I should also recommend Conlon's 'Differentiable Manifolds' (2ed, Birkhauser), as it is the perfect follow-up to Boothby. --A

Very Nice Nontrivial Introduction
This book is a careful treatment of the subjects in the title. It is an introduction, but it manages to cover quite a bit of ground with lots of examples to illustrate. One of it's distinguishing pointsis the way inwhich the concrete, coordinate based calculations are emphasized even whileusually presenting the more abstract, coordinate free approach as well.

The book does a good job at stimulating those studying it to developintuition. I found the book helpful when I was first studying the subject. ... Read more

Product Description
Riemannian geometry has today become a vast and important subject. This new book of Marcel Berger sets out to introduce readers to most of the living topics of the field and convey them quickly to the main results known to date. These results are stated without detailed proofs but the main ideas involved are described and motivated. This enables the reader to obtain a sweeping panoramic view of almost the entirety of the field. However, since a Riemannian manifold is, even initially, a subtle object, appealing to highly non-natural concepts, the first three chapters devote themselves to introducing the various concepts and tools of Riemannian geometry in the most natural and motivating way, following in particular Gauss and Riemann. ... Read more

Customer Reviews (3)

Wonderful overview!

Excellent exposition of a wide variety of topics in geometry for the non-expert like me. It helped me get a better grasp of a number of differential geometric structures that are used in theoretical physics these days, without having to wade through a bunch of proofs. The list of references is immense, so that the reader can follow up with more detailed, and more rigorous, material for topics of choice. Highly recommended for mathematicians and physicists alike.

A superb, rigorous overview of the topic, but without the proofs
This is a beautifully written survey of a wide swath of modern mathematics and mathematical physics related to Riemannian geometry.For a non-expert, like me, it allowed me to get a broad perspective on the field without having to struggle through so many proofs.The explanations are clear and rigorous (two goals which are frequently in conflict).There is a vast bibliography as well.A number of problems I was familiar with in other contexts turn out to have unplumbed (by me at least) geometrical significance. All in all, a wonderful book.

A fine book.
This book is a good reference for anyone who wants to know about the development of riemannian geometry. It covers many active topics in modern differential geometry for every graduated students who can choose some interested fields for the thesis. ... Read more

Product Description
In this book, the global differential geometry ofWeingarten surfaces and hypersurfaces in E4 had beendiscussed. This leads to apply some global methods on a class of Weingarten surfaces and hyper surfaces satisfying umbilical boundary conditions. Also, the characterization of hyperspheres among the umbilical boundary conditions (U.B.C.) Weingarten hypersurfaces with some additionalconditions had been discussed. ... Read more

Product Description

Both classical geometry and modern differential geometry have been active subjects of research throughout the 20th century and lie at the heart of many recent advances in mathematics and physics. The underlying motivating concept for the present book is that it offers readers the elements of a modern geometric culture by means of a whole series of visually appealing unsolved (or recently solved) problems that require the creation of concepts and tools of varying abstraction. Starting with such natural, classical objects as lines, planes, circles, spheres, polygons, polyhedra, curves, surfaces, convex sets, etc., crucial ideas and above all abstract concepts needed for attaining the results are elucidated. These are conceptual notions, each built "above" the preceding and permitting an increase in abstraction, represented metaphorically by Jacob's ladder with its rungs: the 'ladder' in the Old Testament, that angels ascended and descended...

In all this, the aim of the book is to demonstrate to readers the unceasingly renewed spirit of geometry and that even so-called "elementary" geometry is very much alive and at the very heart of the work of numerous contemporary mathematicians. It is also shown that there are innumerable paths yet to be explored and concepts to be created. The book is visually rich and inviting, so that readers may open it at random places and find much pleasure throughout according their own intuitions and inclinations.

Marcel Berger is the author of numerous successful books on geometry, this book once again is addressed to all students and teachers of mathematics with an affinity for geometry.

Product Description
The origins of differential geometry go back to the early days ofthe differential calculus, when one of the fundamental problems was thedetermination of the tangent to a curve. With the development of thecalculus, additional geometric applications were obtained. The principalcontributors in this early period were Leonhard Euler (1707- 1783),GaspardMonge(1746-1818), Joseph Louis Lagrange (1736-1813), andAugustinCauchy (1789-1857). A decisive step forward was taken by KarlFriedrichGauss (1777-1855) with his development of the intrinsic geometryona surface. This idea of Gauss was generalized to n( > 3)-dimensionalspaceby Bernhard Riemann (1826- 1866), thus giving rise to the geometrythat bears his name.This book is designed to introduce differential geometry to beginninggraduate students as well as advanced undergraduate students (this introduction in the latter case is important for remedying the weakness of geometry in the usual undergraduate curriculum). In the last couple of decades differential geometry, along with other branches of mathematics,has been highly developed. In this book we will study only the traditionaltopics, namely, curves and surfaces in a three-dimensional Euclidean spaceE3. Unlike most classical books on the subject, however, more attention ispaid here to the relationships between local and global properties, asopposed to local properties only. Although we restrict our attention tocurves and surfaces in E3, most global theorems for curves and surfaces inthis book can be extended to either higher dimensional spaces or moregeneral curves and surfaces or both. Moreover, geometric interpretationsare given along with analytic expressions. This will enable students tomake use of geometric intuition, which is a precious tool for studyinggeometry and related problems; such a tool is seldom encountered in otherbranches of mathematics. ... Read more

Customer Reviews (1)

Finally, a book on differential geometry for someone who doesn't already know it.
I have been looking for a really long time for a decent introduction to differential geometry.I rate this one 5 stars just because out of all the books I have gotten from my school library (probably about 10 by now), this one is the only one I have seen that doesn't assume that you already are good with all the shorthand notation that is often used in texts on differential geometry.It has been really frustrating to find a simple explanation of the symbols used in most of these books, and none of them, though they claimed to be introductions to the topic, ever really explained themselves adequately.Basically, this book has been the only one so far that has been able to bridge the gap for me between the sort of undergraduate mathematics I am used to and differential geometry.I had previously felt like there was this nearly impenetrable barrier - created by some sort of inner circle that I had to belong to, either by taking a class or having it explained to me by someone who was already familiar with it - that I was just going to have to but my head up against until I broke it down.

But fortunately this book came along, which has made the whole thing very simple for me, and is at just the right level so that when I finish it off, I can go off into those other textbooks that seemed to have been written in a different language and be able to understand what they are saying!

I give this book a 5 star rating for the above reasons, coupled with the fact that this book actually has all the answers (nearly) for the problems in the back - why on earth don't most books just put the darned answers in the back?Otherwise, how do you know if you are doing the problems correctly, I think its rather ridiculous - I surely learn the most when I see that I have done a problem the wrong way, and I go hunting for errors, and usually either find something trivial, or have to rework my entire conception of the idea - which would be impossible without the answers!Either way, this is a good book that really takes you from the basics and gets you into the door on this interesting subject! ... Read more

Product Description
This treatment of differential geometry and the mathematics required for general relativity makes the subject of this book accessible for the first time to anyone familiar with elementary calculus in one variable and with a knowledge of some vector algebra. The emphasis throughout is on the geometry of the mathematics, which is greatly enhanced by the many illustrations presenting figures of three and more dimensions as closely as book form will allow. The imaginative text is a major contribution to expounding the subject of differential geometry as applied to studies in relativity, and will prove of interest to a large number of mathematicians and physicists. Review from L'Enseignement Mathématique ... Read more

Customer Reviews (7)

Excellent, BUT...
This book is excellent, but I can't understand that this second edition contains so many "typographical" errors! So, the novice reader will have to consult reference material to make sure he doesn't overlook something, which somehow defeats the purpose of the book.

Excellent Introduction to Semi-Riemannian Geometry
The authors of this excellent text include a memorable passage in the Introduction that perfectly captures the purpose and primary strength of the book:

"The title of this book is misleading.Any possible title would mislead somebody.'Tensor Analysis' suggests to a mathematician an ungeometric, manipulative debauch of indices, with tensors ill-defined as 'quantities that transform according to' unspeakable formulae.'Differential Geometry' would leave many a physicist unaware that the book is about matters with which he is very much concerned.We hope that 'Tensor Geometry' will at least lure both groups to look more closely."

Dodson and Poston's text is a welcome entry in that all-too-small class of books that attempt to bridge the conceptual gulf that separates mathematicians from physicists when they write about differential geometry and general relativity.Modern mathematical treatments of both Riemannian and Lorentzian geometry are typically written primarily in concise and conceptually rich coordinate-free notation;physicists, in sharp contrast, tend to write almost exclusively in a notation that stresses the use of local coordinate systems and index manipulation.A person who is educated in one of these traditions must apply himself with diligence to become proficient in the other;however, this "bilingual" proficiency is surely necessary for the serious students of general relativity, who must study literature written in both styles.

Dodson and Poston's book provides an accessible introduction to the mathematics of general relativity, and it should be particularly useful to both mathematicians and physicists as they develop their abilities to read and write in both coordinate-free and index-based notations.The book is written at a level that should make it accessible to anyone who has studied multi-variable calculus and linear algebra.It is not a complete introduction to either modern differential geometry or general relativity, nor do the authors claim that it is.After all, Spivak devoted five volumes to Riemannian geometry alone and still failed to provide an exhaustive introduction; the subject is enormous in scope.

Mathematicians who find this book helpful in their studies of general relativity might consider looking into the following books, each of which is written in the same mathematical style:(1)Gravitational Curvature by Theodore Frankel (offers a beautiful derivation of the Raychaudhuri Equation);(2)Manifolds, Tensor Analysis and Applications by Abraham, Marsden and Ratiu (Chapters 6, 7 and 8 offer an exceptionally lucid introduction to differential forms, integration on manifolds, the Hodge star operator, the codifferential, and applications of these materials to physics);(3)The Geometry of Kerr Black Holes by Barrett O'Neill (if you want to UNDERSTAND the use of the Weyl curvature tensor in defining the Petrov Type of a spacetime, then read Chapter 5 of this wonderful book);(4)Semi-Riemannian Geometry with Applications to Relativity by Barrett O'Neill (makes an excellent companion text to Dodson and Poston as a mathematically rigorous introduction to GR);(5)General Relativity for Mathematicians by Rainer Sachs and Hung-Hsi Wu(a masterpiece, difficult to find today but worth the effort).For a more far-ranging treatment of geometry with applications beyond GR, Theodore Frankel's The Geometry of Physics is also highly recommended.

After one has used Dodson and Poston and some of these other references as a sort of "Rosetta Stone," then one can become reasonably proficient in deciphering both coordinate-free and coordinate-based literature and translating one into the other. It is sad that the educational process is necessarily so inefficient, but we must be grateful for books like Dodson and Poston's that help us in the endeavor.

A must for mathematicians interested in cosmology
Besides providing in clear-cut fashion the mathematics essential to research in cosmology, the authors simplify many concepts the physicists make opaque.

Excellent, But Has Flawed Editing
I found this book much more comprehensible than the Bishop & Goldberg text (perhaps because the specifics of my particular graduate program made this approach more accessible). The only problem is that in some important areas of the text there are typographical errors; this text needs a new corrected edition--otherwise I have no complaints. The only recommendation I would make to a prospective reader is to obtain a copy of the excellent, old, out-of-print (but still readily avilable used, on the web) Vector and Tensor Analysis by Louis Brand and master the tensor chapter in it. This will prepare the reader to gain an idea of where the authors are heading with their modern, abstract functional analysis approach.
The reader will be greatly assisted by a solid understanding of linear algebra and a preparatory course in functional analysis wouldn't hurt, either. This material is challenging even for a math graduate student; any high school student who could master this book would have to be gifted, indeed.

Good introduction
I agree with previous reviewers, and only wish to add a few comments: 1. This book assumes very little on the part of the reader, which makes it ideal for beginners, as long as they're mature readers. 2. Like many books out there, everything in this book is real and finite dimensional, which is a bit disappointing. 3. It's not as advanced as the writers or reviewers would like to think. For instance, no differential forms, no killing vectors, and although there's a chapter on lie groups it treats only their geometrical aspects and not the algebraic ones.4. However, it contains two (extensive) chapters on SR and GR which are pure gold, I say! Everything is done from the geometricalpoint of view, and only AFTER all of the math has been introduced, so the discussion is mature and elegant.In short, this is a good book to read for the geometrical intuition but don't count on it to explain everything about differential geometry. Enjoy! ... Read more

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This textbook for second-year graduate students is intended as an introduction to differential geometry with principal emphasis on Riemannian geometry. Chapter I explains basic definitions and gives the proofs of the important theorems of Whitney and Sard. Chapter II deals with vector fields and differential forms. Chapter III addresses integration of vector fields and $p$-plane fields. Chapter IV develops the notion of connection on a Riemannian manifold considered as a means to define parallel transport on the manifold. The author also discusses related notions of torsion and curvature, and gives a working knowledge of the covariant derivative. Chapter V specializes on Riemannian manifolds by deducing global properties from local properties of curvature, the final goal being to determine the manifold completely. Chapter VI explores some problems in PDEs suggested by the geometry of manifolds.

The author is well-known for his significant contributions to the field of geometry and PDEs--particularly for his work on the Yamabe problem--and for his expository accounts on the subject. ... Read more

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This is the first volume of a three-volume introduction to modern geometry, with emphasis on applications to other areas of mathematics and theoretical physics. Topics covered include tensors and their differential calculus, the calculus of variations in one and several dimensions, and geometric field theory. This material is explained in as simple and concrete a language as possible, in a terminology acceptable to physicists. The text for the second edition has been substantially revised. ... Read more

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Written for the physicist in mind
This book, written by some of the master expositors of modern mathematics, is an introduction to modern differential geometry with emphasis on concrete examples and concepts, and it is also targeted to a physics audience. Each topic is motivated with examples that help the reader appreciate the essentials of the subject, but rigor is not sacrificed in the book.

In the first chapter the reader gets a taste of differentiable manifolds and Lie groups, the later gving rise to a discussion of Lie algebras by considering, as usual, the tangent space at the identity of the Lie group. Projective space is shown to be a manifold and the transition functions explicitly written down. The authors give a neat example of a Lie group that is not a matrix group. A rather quick introduction to complex manifolds and Riemann surfaces is given, perhaps too quick for the reader requiring more details. Homogeneous and symmetric spaces are also discussed, and the authors plunge right into the theory of vector bundles on manifolds. Thus there is a lot packed into this chapter, and the authors should have considered spreading out the discussion more, as it leaves the reader wanting for more detail.

The authors consider more fundamental questions in smooth manifolds in chapter 3, with partitions of unity used to prove the existence of Riemannian metrics and connections on manifolds. They also prove Stokes formula, and prove the existence of a smooth embedding of any compact manifold into Euclidean space of dimension 2n + 1. Properties of smooth maps, such as the ability to approximate a continuous mapping by a smooth mapping, are also discussed. A proof of Sard's theorem is given, thus enabling the study of singularities of a mapping. The reader does get a taste of Morse theory here also, along with transversality, and thus a look at some elementary notions of differential topology. An interesting discussion is given on how to obtain Morse functions on smooth manifolds by using focal points.

Notions of homotopy are introduced in chapter 3, along with more concepts from differential topology, such as the degree of a map. A very interesting discussion is given on the relation between the Whitney number of a plane closed curve and the degree of the Gauss map. This leads to a proof of the important Gauss-Bonnet theorem. Degree theory is also applied to vector fields and then to an application for differential equations, namely the Poincare-Bendixson theorem. The index theory of vector fields is also shown to lead to the Hopf result on the Euler characteristic of a closed orientable surface and to the Brouwer fixed-point theorem.

Chapter 4 considers the orientability of manifolds, with the authors showing how orientation can be transported along a path, thus giving a non-traditional characterization as to when a connected manifold is orientable, namely if this transport around any closed path preserves the orientation class. More homotopy theory, via the fundamental group, is also discussed, with a few examples being computed and the connection of the fundamental group with orientability. It is shown that the fundamental group of a non-orientable manifold is homomorphic onto the cyclic group of order 2. Fiber bundles with discrete fiber, also known as covering spaces, are also discussed, along with their connections to the theory of Riemann surfaces via branched coverings. The authors show the utility of covering maps in the calculation of the fundamental group, and use this connection to introduce homology groups. A very detailed discussion of the action of the discrete group on the Lobachevskian plane is given.

Absolute and relative homotopy groups are introduced in chapter 5,and many examples are given of their calculation. The idea of a covering homotopy leads to a discussion of fiber spaces. The most interesting discussion in this chapter is the one on Whitehead multiplication, as this is usually not covered in introductory books such as this one, and since it has become important in physics applications. The authors do take a stab at the problem of computing homotopy groups of spheres, and the discussion is a bit unorthodox since it depends on using framed normal bundles.

The theory of smooth fiber bundles is considered in the next chapter. The physicist reader should pay close attention to this chapter is it gives many insights into the homotopy theory of fiber bundles that cannot be found in the usual books on the subject. The discussion of the classification theory of fiber bundles is very dense but worth the time reading. Interestingly, the authors include a discussion of the Picard-Lefschetz formula, as an example of a class of "fiber bundles with singularities". Those interested in the geometry of gauge field theories will appreciate the discussion on the differential geometry of fiber bundles.

Dynamical systems are introduced in chapter 7, first as defined over manifolds, and then in the context of symplectic manifolds via Hamaltonian mechanics. Liouville's theorem is proven, and a few examples are given from relativistic point mechanics. The theory of foliations is also discussed, although the discussion is too brief to be of much use. The authors also consider variational problems, and given its importance in physics, they continue the treatment in the last chapter of the book, giving several examples in general relativity, and in gauge theory via a consideration of the vacuum solutions of the Yang-Mills equation. The physicist reader will appreciate this discussion of the classical theory of gauge fields, as it is good preparation for further reading on instantons and the eventual quantization of gauge fields.

A masterful sequel!
Novikov et al's first volume was the defining book on differential geometry (S-V 93). The second volume picks up on the detailed theory of manifolds and topology and other advanced theories of differentialgeometry, including homotopy groups, Lie algebras and digressing intophysical theories as well (eg.Yang-Mills) giving one of the juciest bookson the subject - an utter delight! ... Read more

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Geometry, this very ancient field of study of mathematics, frequently remains too little familiar to students. Michèle Audin, professor at the university of Strasbourg has written a book allowing them to remedy this situation and, and starting from linear algebra extend their knowledge of affine, euclidian and projective geometry, conic and quadric sections, curves and surfaces. It includes many nice theorems like the nine-point circle, Feuerbach's theorem, and so on. Everything is presented clearly and rigourously. Each property is proved, examples and exercises illustrate the course content perfectly. Precise hints for each exercise are provided at the end of the book. This very comprehensive text is addressed to students at upper undergraduate and Master's level to discover geometry and deepen their knowledge and understanding. ... Read more

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Math
Un libro con un contenido propicio para lo que buscaba, el lenguaje se entiende claramente y no deja espacio a dudas ... Read more

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This book, based on a graduate course on Riemannian geometry and analysis on manifolds, held in Paris, covers the topics of differential manifolds, Riemannian metrics, connections, geodesics and curvature, with special emphasis on the intrinsic features of the subject. Classical results on the relations between curvature and topology are treated in detail. The book is quite self-contained, assuming of the reader only differential calculus in Euclidean space. It contains numerous exercises with full solutions and a series of detailed examples which are picked up repeatedly to illustrate each new definition or property introduced.

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Hard to say
This was the official 100% recommended, guaranteed text for my Riemannian Geometry class.Supplementing this book with do Carmo's text, I was able to get something out of the class, but I think rereading both of them now would be much better.The condensed one chapter course on manifolds at the beginning of GHL wasn't sufficient to learn/relearn everything I needed to know in order to read it for the first time. ... Read more

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In this volume the authors seek to illustrate how methods of differential geometry find application in the study of the topology of differential manifolds. Prerequisites are few since the authors take pains to set out the theory of differential forms and the algebra required. The reader is introduced to De Rham cohomology, and explicit and detailed calculations are present as examples. Topics covered include Mayer-Vietoris exact sequences, relative cohomology, Pioncare duality and Lefschetz's theorem. This book will be suitable for graduate students taking courses in algebraic topology and in differential topology. Mathematicians studying relativity and mathematical physics will find this an invaluable introduction to the techniques of differential geometry. ... Read more

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"Geometry and Physics" addresses mathematicians wanting to understand modern physics, and physicists wanting to learn geometry. It gives an introduction to modern quantum field theory and related areas of theoretical high-energy physics from the perspective of Riemannian geometry, and an introduction to modern geometry as needed and utilized in modern physics. Jürgen Jost, a well-known research mathematician and advanced textbook author, also develops important geometric concepts and methods that can be used for the structures of physics. In particular, he discusses the Lagrangians of the standard model and its supersymmetric extensions from a geometric perspective.

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This book offers an introduction to the theory of differentiable manifolds and fiber bundles. It examines bundles from the point of view of metric differential geometry: Euclidean bundles, Riemannian connections, curvature, and Chern-Weil theory are discussed, including the Pontrjagin, Euler, and Chern characteristic classes of a vector bundle. These concepts are illustrated in detail for bundles over spheres.

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This is an introduction to noncommutative geometry, with special emphasis on those cases where the structure algebra, which defines the geometry, is an algebra of matrices over the complex numbers. Applications to elementary particle physics are also discussed. This second edition is thoroughly revised and includes new material on reality conditions and linear connections plus examples from Jordanian deformations and quantum Euclidean spaces. Only some familiarity with ordinary differential geometry and the theory of fiber bundles is assumed, making this book accessible to graduate students and newcomers to this field. ... Read more

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Very nice, lots of good stuff
This book is (partially) the answer to my prays: an introductory book on noncommutative geometry, something I've been waiting since I discovered thetopic in Connes' seminal text, which I've also reviewed here. Instead ofexposing the historical origins, then firing a goddamn chaingun of advancedtopics(something quite fascinating, because of the potential of thetheory,but not pedagogical), Madore uses a more friendly way of exposingthings,by mantaining a compromise between the most natural motivations tothetechniques of the subject and the places where the background neededis not so overwhelming. He do teach much of the background (in the sensethatyou don't need to master functional analysis, operator algebras andadvanced differential geometry), but he goes quite fast on it, requiring arather maturemathematical mind. As noncommutative geometry is not for thefaint of theheart, I guess he's not asking too much after all.

Thepedagogy of the book is also benefitted from the post-"Connes'book" evolution of noncommutative geometry, because in 1999 the theoryand its (real and potential) applications were a great deal more mature andsolid than in 1994. Being this theory a work in progress, the better themathknowledge the reader has, the more he or she will learn from Madore'sbook, which stands maybe as the only pedagogical exposition ofnoncommutative geometry (now I'm waiting for the huge book fromGarcia-Bondia and his colaborators, to be published by Birkhauser in 2001,hope that it contains more background; it would be very useful for thoseinterested in beginning research on the area).

An Introduction to Noncommutative Differential Geometry and
FOR PHYSICIST, I strongly reccomend this book! There are so many physical examples in this book. Always we physicists hate mathamatical proofs likea torture. But this book concentrates applications to physics. If you wantto study Noncommutative Geometry as a physicist, this book should be chosenas the first introduction! ... Read more