Book Description

This text is an introduction to the theory of differentiable manifolds and fiber bundles. The only requisites are a solid background in calculus and linear algebra, together with some basic point-set topology. The first chapter provides a comprehensive overview of differentiable manifolds. The following two chapters are devoted to fiber bundles and homotopy theory of fibrations. Vector bundles have been emphasized, although principal bundles are also discussed in detail. The last three chapters study 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. Chapter 5, with its focus on the tangent bundle, also serves as a basic introduction to Riemannian geometry in the large. This book can be used for a one-semester course on manifolds or bundles, or a two-semester course in differential geometry.

Book Description

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.

Customer Reviews (1)

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

Book Description
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

Customer Reviews (2)

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

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

Book Description
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, starting from linear algebra, extend their knowledge of affine, Euclidean and projective geometry, conic sections and quadrics, 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 most of the exercises 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

Customer Reviews (1)

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

Book 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

Book Description
This is an introduction to geometrical topics useful in applied mathematics and theoretical physics, including manifolds, metrics, connections, Lie groups, spinors and bundles, preparing the reader for the study of modern treatments of mechanics, gauge field theories, relativity and gravitation. ... Read more

Customer Reviews (1)

Highly recommended textbook
I think the book by Crampin and Pirani may serve as an example of a thoughtfully written and useful textbook.
It treats those parts of differential geometry which are important in application (as the title indicates), especially in physics and related subjects.
It is written in a clear and comprehensible style and may also be used by beginners not being exposed to lectures on the subject.
Perhaps it should be mentioned that the book does not contain integration theory of forms.
However, I do not regard this as a substantial drawback, there are manystandard sources (Flanders, Spivak, ...) to fill this gap after becoming familiar with the rich stuff which IS there.
(No book on the subject gives you all you might want and the one under consideration gives you quite a lot.)
... Read more

Book Description
Since the early work of Gauss and Riemann, differential geometry has grown into a vast network of ideas and approaches, encompassing local considerations such as differential invariants and jets as well as global ideas, such as Morse theory and characteristic classes. In this volume of the Encyclopaedia, the authors give a tour of the principal areas and methods of modern differential geomerty. The book is structured so that the reader may choose parts of the text to read and still take away a completed picture of some area of differential geometry. Beginning at the introductory level with curves in Euclidian space, the sections become more challenging, arriving finally at the advanced topics which form the greatest part of the book: transformation groups, the geometry of differential equations, geometric structures, the equivalence problem, the geometry of elliptic operators. Several of the topics are approaches which are now enjoying a resurgence, e.g. G-structures and contact geometry. As an overview of the major current methods of differential geometry, EMS 28 is a map of these different ideas which explains the interesting points at every stop. The authors' intention is that the reader should gain a new understanding of geometry from the process of reading this survey. ... Read more

Book Description
This book is a translation of an authoritative introductory text based on a lecture series delivered by the renowned differential geometer, Professor S S Chern in Beijing University in 1980. The original Chinese text, authored by Professor Chern and Professor Wei-Huan Chen, was a unique contribution to the mathematics literature, combining simplicity and economy of approach with depth of contents. The present translation is aimed at a wide audience, including (but not limited to) advanced undergraduate and graduate students in mathematics, as well as physicists interested in the diverse applications of differential geometry to physics. In addition to a thorough treatment of the fundamentals of manifold theory, exterior algebra, the exterior calculus, connections on fiber bundles, Riemannian geometry, Lie groups and moving frames, and complex manifolds (with a succinct introduction to the theory of Chern classes), and an appendix on the relationship between differential geometry and theoretical physics, this book includes a new chapter on Finsler geometry and a new appendix on the history and recent developments of differential geometry, the latter prepared specially for this edition by Professor Chern to bring the text into perspectives. ... Read more

Customer Reviews (6)

Great Summary
Let me begin by saying that I am biased.I worked as Mr. Chern's assistant in a differential geometry class when I was a grad student.He was a great person to work for and his lectures were well organized.This book is a NOT aimed at the typical undergraduate.It is a major advance in comprehensability from the books from which I learned the covered material.Modern differential geometry does not yet have a great, easy for the novice, self-study friendly text that really covers the material - this book and the Russian trilogy by Dubrovin, et al. are major steps along the way.

an excellentbook!
As many professors in China recommend, itis an excellent book by a great Geometrician. Though it may not be a beginning book, it should appear on your shelf as a classic one!

Be Careful!
This book is written by famous authors alright. It may have their reason in the way they choose those materials and the way they are presented.

The point is: as an introductory text, the various ideas and structures are not well motivated. They may be economical in the way of the presentation. However, it never seems natural from the point of view of a beginner. It is more natural to start with Riemannian geometry and then proceed to the more general concept of vector bundles and connections. It is in Riemannian geometry, that it is natural to first introduce the concept of a geodesic, and this leads, though a lot of books dont do it this way, to the concept of Levi -Civita connection and therefore holonomy and curvature. The general concept of vector bundles and connections before introducing the Riemannian geometry, makes a complex subject even more abstract and though maybe economical from the point of view of the writers, are formidable for a reader.

Even the presentation of specific facts, the book should emphassize, for the benefit of the reader, the structrual (pictorial) aspects more than it does, to illuminate the essence of the formulas, for example, the way it introduces the theta forms on frame bundle omits entirely in mentioning that the essence of thse forms is simply the concept of a coframe. It merely constructs these forms using local coordinates, which seems to be quite tricky to get to its bottom.

To readers of this book
I am reading this book now. It is as the other reviewers said,
rather condensed. However, it would not be beyond comprehension
if the crucial pictures are established. It is my personal opinion that the first crucial place where it should be understood without any compromise is the section on the frame bundle. Later chapters build on this. Previous chapters are
synthesized here. To any readers who are interested, you are invited to discuss this book. My email address is topollogy@hotmail.com (Notice there are two "l" in "topollogy")

dense book
This book contains most important material in differential geometry in about 330 page. No exercise, few exaples make this book very dense, which is just the style of Chinese professors.
It deserves cautions that in chapter 8, Chern introduce the connection proposed by himself in 1948, which is the proper tools for finsler geometry. ... Read more

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

Customer Reviews (1)

Good Concise Coverage of Synthetic Projective Geometry
Coxeter provides good coverage of the fundamental concepts of synthetic projective geometry. This is a good introductory book. The examples provided are clear.Coxeter's style is concise. Topics such as Desargues theorem , Pappus's theorem and conics are covered. The solutions to some exercises can be found in the back of the book. The book addresses finite geometries and analytic projective with a chapter on each. ... Read more

Book Description
This book not only explains and develops the classical theory of curves, but also allows the reader to reproduce and study curves and surfaces using computer methods. This second edition contains eight new chapters on global curve theory, space curves, minimal surfaces, inversions, cyclides, the Gauss-Bonnet Theorem and global surface theory. ... Read more

Customer Reviews (7)

Brilliant book
The book I have is the 3d Ed, which is greatly improved (Mathematica code is left at the end of each chapter and one could download all program files from the publisher's website). Together with much better clarity of presentation, this extended edition gives excellent overview of various topics supported by more than 300 illustrations that help to develope the intuitive feel for geometry in general. I would highly recommend this book for students who needs a comprehensive approach with lots of examples one could try out using Mathematica. Learning new concepts and visualizing them with Mathematica is great fun! The beauty of geometry is more accessible than ever before.

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

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

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

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

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

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

Book Description
Introduction to Differential Geometry with applications to Navier-Stokes Dynamics is an invaluable manuscript for anyone who wants to understand and use exterior calculus and differential geometry, the modern approach to calculus and geometry.

Author Troy Story makes use of over thirty years of research experience to provide a smooth transition from conventional calculus to exterior calculus and differential geometry, assuming only a knowledge of conventional calculus.Introduction to Differential Geometry with applications to Navier-Stokes Dynamics includes the topics:

• Geometry,
• Exterior calculus,
• Homology and co-homology,
• Applications of differential geometry and exterior calculus to: Hamiltonian mechanics, geometric optics, irreversible thermodynamics, black hole dynamics, electromagnetism, classical string fields, and Navier-Stokes dynamics.Download Description
  Introduction to Differential Geometry with applications to Navier-Stokes Dynamics is an invaluable manuscript for anyone who wants to understand and use exterior calculus and differential geometry, the modern approach to calculus and geometry.

  Author Troy Story makes use of over thirty years of research experience to provide a smooth transition from conventional calculus to exterior calculus and differential geometry, assuming only a knowledge of conventional calculus.Introduction to Differential Geometry with applications to Navier-Stokes Dynamics includes the topics:
  

  • Geometry
  • Exterior calculus
  • Homology and co-homology
  • Applications of differential geometry and exterior calculus to: Hamiltonian mechanics, geometric optics, irreversible thermodynamics, black hole dynamics, electromagnetism, classical string fields, and Navier-Stokes dynamics. ... Read more

    Customer Reviews (3)

    Excellent Investment
    Beginning with a definiton of Euclidean geometry, this book presents a logical and rigorous introduction to differential geometry and exterior calculus. Rather than using the style of of writing theorems followed by proofs or sketches of proofs, the writer displays fundamental definitions followed by examples and a chapter on applications to dynamics. In contrast to one of the reviewers, my purchase of this book is an excellent investment.

    Insane
    Refering to this book as an introduction to anything is insane. Definitions occur rarely and invariably use terms which are themselves undefined.While it is reasonable for the author of a book on differentiable geometry to not define the derivative of real valued function of a single real variable, or a jacobian matrix, a notion such as a metric tensor should be fully defined.
    This book lacks any pretense of rigour (I do not recall seeing the word proof once!). This alone is not a damning feature but it is also lacking in tools to develop intuition as well. Do not expect to be able to prove or compute anything after reading this book. I got the e-book version and it is the worst 6 dollars I have ever spent.

    Highly recomended
    This book has the clearest and most direct introduction to differential geometry that I've seen, thus, I highly recommend it.
    
    One of the book's highlights is chapter 2, which presents a discussion of exterior calculus assuming only a knowledge of conventional calculus. Unlike most books on this topic(exterior calculus), this book includes a definition of the exterior derivative rather than a few examples.
    
    Another highlight is the chapter on dynamics, where it is shown that many areas of dynamics can be described by differential one-forms, including Navier-Stokes dynamics for incompressible fluids.
    
    
    
    ... Read more

Book 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 (1)

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

Book Description

Customer Reviews (1)

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

Book Description
This book is an introduction to modern differential geometry. The authors begin with the necessary tools from analysis and topology, including Sard's theorem, de Rham cohomology, calculus on manifolds, and a degree theory. The general theory is illustrated and expanded using the examples of curves and surfaces. In particular, the book contains the classical local and global theory of surfaces, including the fundamental forms, curvature, the Gauss-Bonnet formula, geodesics, and minimal surfaces. ... Read more

Book Description
This classic work is now available in an unabridged paperback edition. Stoker makes this fertile branch of mathematics accessible to the nonspecialist by the use of three different notations: vector algebra and calculus, tensor calculus, and the notation devised by Cartan, which employs invariant differential forms as elements in an algebra due to Grassman, combined with an operation called exterior differentiation. Assumed are a passing acquaintance with linear algebra and the basic elements of analysis. ... Read more

Book Description
This book is an exposition of semi-Riemannian geometry (also called pseudo-Riemannian geometry)--the study of a smooth manifold furnished with a metric tensor of arbitrary signature. The principal special cases are Riemannian geometry, where the metric is positive definite, and Lorentz geometry. For many years these two geometries have developed almost independently: Riemannian geometry reformulated in coordinate-free fashion and directed toward global problems, Lorentz geometry in classical tensor notation devoted to general relativity. More recently, this divergence has been reversed as physicists, turning increasingly toward invariant methods, have produced results of compelling mathematical interest. ... Read more

Customer Reviews (6)

The Best Introduction to General Relativity
If you want to engage in a serious study of general relativity, then you must master the mathematical language of semi-Riemannian manifolds in which it is cast. Sadly, the development of classical Riemannian geometry as studied by pure mathematicians only parallels the development of semi-Riemannian geometry in the early stages;eventually, the two subjects diverge rather drastically.For example, the famous Hopf-Rinow Theorem, one of the cornerstones of modern Riemannian geometry, simply has no Lorentzian analogue at all;every single equivalence in the theorem fails in Lorentzian geometry.Thus, one could master all five volumes of Spivak's definitive treatment of Riemannian geometry and still be unprepared to deal with light cones, timelike, null and spacelike geodesics, and the multitude of other uniquely semi-Riemannian constructs that appear in general relativity.O'Neill's wonderful book, which first appeared in 1983, provides the well-prepared reader with a mathematically rigorous, thorough introduction to both Riemannian and semi-Riemannian geometry, showing how they are similar and pointing out clearly where they differ.After developing the mathematical machinery in the early chapters, the last part of the book turns to general relativity by offering lucid introductions to the Robertson-Walker cosmological models (Big Bang singularities), the Schwarzschild model for a single non-rotating star (including black holes), and a brief introduction to Penrose-Hawking causality theory.If you would like to study a pure Riemannian text in parallel with this one, I would recommend the text by Boothby, written at a comparable level of difficulty, which remains one of the clearest and most accessible Riemannian geometry texts on the market.For the serious reader who wishes to continue on with a study of the Kerr solution to Einstein's equations, modeling the exterior spacetime of a rotating star, O'Neill wrote an entire book on the subject in 1995, now difficult to find but well worth tracking down.This 1995 text contains one of the clearest, most accessible
introductions available to the difficult subjects of the algebraic classification of the Weyl curvature tensor and the corresponding Petrov classification of spacetimes.I studied from O'Neill's 1983 text when it first came out and I have continued to use it as the primary text for an advanced undergraduate course I have taught over the past 20 years.It is not an "easy" text to read, but then, I have never found the "easy" introduction to differential geometry and general relativity.The reviewer who says this is not a suitable first text is simply in error;there is no better first text on the subject.If you have studied linear algebra, advanced calculus, and a little topology, then with dedication and hard work, you can learn more from O'Neill's text than from many of the far more popular recent texts, written by physicists, which attempt to circumvent the mathematics insofar as is possible while introducing general relativity.This is a perilous course for which the serious student will pay dearly later on, when she/he wants to study any of the many areas of modern physics in which differential geometry (differential forms, bundle theory, connections on a principle fiber bundle, gauge theory, etc.) plays an essential role.

Great book, terrible print quality
This is a wonderful book, with a clear, concise and precise exposition of the fundamental idea in riemannian and semi-riemannian geometry.Although I would not recommend it as a first text, it will be the text that you continue to reference later, and turn to when you want the best mathematical treatment.

However, I do not recommend that you buy a new copy.The print quality is terrible; the binding is poor, but even worse, the text quality is absurd.I have been using a library copy with cloth binding and sharp, clear text.It is obvious that the new printing in the green cover is based on a photocopy of the original rather than a new typesetting.While this means that no errors have been introduced, I found it painful to read.I would suggest looking for a used copy.

So 5 stars for the book, but only 3 stars for this printing.

Very good contents but..
The only drawback, and it is a serious one, is the binding. For a such expensive book, one could expect a DECENT binding, but the outcome is a SHAME.

So 5 star for the contents an 0 for the binding

Addendum
This book is now available at Amazon.co.uk!

Its contents are: Manifold Theory. Tensors. Semi-Riemannian Manifolds. Semi-Riemannian Submanifolds. Riemannian and Lorentz Geometry. Special Relativity. Constructions.Symmetry and Constant Curvature. Isometries. Calculus of Variations.Homogeneous and Symmetric Spaces. General Relativity; Cosmology.Schwarzschild Geometry. Causality in Lorentz Manifolds.

Let's go buy it!

Excellent for beginner and experienced mathematicians
This is one of the best books on Differential Geometry I've ever read. It includes a clear exposition of all the basic results and then goes on to the most deep aspects of the subject, making it useful for undergraduateand graduate students, as well as experienced working mathematicians. It'sa pitty that it's no longer available. ... Read more

Book Description
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