Book Description
Spacetime and Geometry: An Introduction to General Relativity provides a lucid and thoroughly modern introduction to general relativity. With an accessible and lively writing style, it introduces modern techniques to what can often be a formal and intimidating subject.Readers are led from the physics of flat spacetime (special relativity), through the intricacies of differential geometry and Einstein's equations, and on to exciting applications such as black holes, gravitational radiation, and cosmology.For advanced undergraduates and graduate students, or anyone interested in astronomy, cosmology, physics, or general relativity. ... Read more

Customer Reviews (11)

Wordy and Wonderful
This is an advanced text, but all the same it is not particularly rigorous or dense, so it is in principle accessible to the beginner.With an easy authority, Carroll leads us on a wandering journey through the mystical lands of general relativity.This is very different from, and compliments nicely, the clarity and directness of Wald.As a student of GR, I use Wald for the bottom line on any subject, and Carroll for the random physical or computational insights that I invariably find in any section of the book.Carroll's prose is like music to the ear and I always enjoy myself when I decide to open up this book.

Be warned that there are lots of mistakes in this first edition--you might want to wait for the second one.

Also, his chapter on cosmology is better than any I've seen.

BY FAR the best book on GR
I am currently on the 4th chapter of Carroll's "Spacetime and Geometry" and thus far I am amazed at how clear it is.Sure there is a lot of math in it however that also is very clearly explained.In fact, I think that Carroll explains the differential geometry material better than any mathematician has in any book on the subject.If you want to learn general relativity, there is no getting around the math; sooner or later you'll have to learn it.I'd suggest, especially if you are self-studying the subject, to rather pick up this book and go through it than pick up a more "elementary" text and a book on Riemannian geometry to look at later.

(Although I do also highly recommend Kay's (Schaum outline) "Tensor Calculus" for self study.The prima donnas don't like Kay's book because it "doesn't have enough theory."I suppose if a freshman calculus book does not have the Lebesgue integral defined in ti they'll complain about that too.)

Because, you can always skip through certain sections if the math is too heavy and go back through it later.And like I wrote earlier, you won't find a better introduction to the mathematical material than here.

Carroll should be given the Nobel prize for this book.If not in Physics, then in literature.I'd give this textbook 10 stars if I could.

A nice blend of the ideas of physics with mathematics
Kudos to Carroll.

This book is an excellent INTRODUCTION to SR and GR for the graduate physics student as well as the graduate mathematics students.

Pure mathematics often loses sight of the ideas which motivated it and physics often loses the mathematical foundations from which it is built.

This book offers some level of mathematical formalism to the physics student while exposing the ideas motivating the mathematical concepts.

I particularly like how he builds up the mathematical machinery of GR by introducing sets then topology on this set giving a topological space. Now he adds in the ideas of a manifold which make this topological space look like Rn locally with the patches sewn together smoothly. The manifold comes equipped with tangent space, cotangent spaces and their product spaces giving tensor spaces. These are defined nicely with reference to component formalism as well as the multilinear algebra approach as maps from products spaces to the reals, etc. He delves into forms and tantalized the reader with deRham cohomology although doesnt go into it. He shows how these can be differentiated ( exterior derivative ) and integrated.

Now the metric is introduced giving a geometry. To this is added a connection which is independent of the metric and leads to notions of parallel transport and differentiation of tensors ( covariant derivative ). One sees that in a special case one can derive a unique connection from the metric ( Levi-Cevita ) which is used in GR.

Fibre bundles, Lie derivatives, pullbacks etc are introduced as needed.

He then presents some introductory GR material by applying the mathematics.

Great Book But Won't Get You To The Promised Land
My comments come with a few caveats.

1. This is my fourth GR book.
2. I'm not hardcore into physics.I'm not a physic grad and I'm reading GR for fun.I have a decent graduate math background but I've been corrupted with 10+ years in working in various roles software engineering, electronics engineering and marketing.
3. I assume that since you're considering buying this book, you're goal is to get at the "real" GR, not the watered down discover channel version.

With these caveats in mind, here are my comments.

First, on a scale of 1-5, I rank Carroll at level 3 in terms of math/physics maturity and thoroughness.Here is my full ranking of authors from my limited reading:1. schutz2. hartle3. penrose3. carroll 4. wald5. physics journal articles

Second, using the rankings above, I recommend Carroll as the second port of entry.If you're comfortable with multivariable calculus, start with schutz (#1). You'll get warm fuzzies doing the toy exercises. But Schutz is tensor/math-lite.If you've had advanced calculus and geometry already, jump in with carroll (#3).But you'll be hard-pressed to find anyone else as polite to the reader.He won't prepare you for 80 percent of what's published.If you're ready to throw off the training wheels and jump dive into mainstream GR go with Wald (#4).

Note that Hartle (#2) is a good "tweener" book with feel-good exercises and some of the full-on GR equations at the end.I bet most instructors teaching a first year grad course would go with Hartle along with a dose of supplementary material.

Third, don't expect Carroll to be your last GR book purchase if you want to reach the promised land (see caveat #4).Living and breathing GR is found in physics journals and for that you'll need Wald or another advanced GR book.

good math chapters, not at beginner's level after that

I had a course based on that book and I've read chapters 1-6 (out of 9 chapters total) plus all the appendices. Also, I've solved some of the problems.

Please keep in mind my review is from a beginner point of veiw. Readers more experienced in GR may feel different but that book is supposedly written for beginners right?

The math chapters 2 and 3 are worth reading because they will teach you tensor analysis on manifolds in much clearer way than other books. The book makes a clear distinction between assumptions, choices (like working with a metric compatible connection), or derived facts. It is nice that the book makes a difference between a Christoffel connection and a generic connection. The appendices are worth reading too cause they will give you a feeling for some new to you math necessary for GR like pullbacks, Lie Derivatives, hypersurfaces etc.

Chapter 4 is worth reading too cause it makes clear that Einstein's equations are just the simplest guess out of many other possibilities. Also it shows how we generalize physical laws from special relativity to GR making it clear our choices are the simplest ones but not the only ones possible.

The chapters after that discuss applications of GR like black holes, gravitational radiation, cosmology etc. Of these, I've read only the black holes chapters 5 and 6 and I wasn't able to understand 100% what was goin on. The problem was that the book uses concepts that you still don't quite understand if you are a beginner like 'spacelike singularity' or 'conformal diagrams'. That is informative but the book doesn't provide the necessary level of detail and examples for beginners so you could really master such concepts and use them in your practise.

There are problems after each chapter but not the necessary beginners problems that increase your conceptual understanding of the theory. Instead, some of the problems are just tedious algebra of type 'find the curvature for some general form of the metric' for which specialists in the field use symbolic programs like Mathematica. Solving these by hand proves that you can take derivatives and you are a mazochist but not that you understand GR. Other problems are really relevant to your education but are not dirrectly connected to the discussion in the text. Because of that you have to solve them from scratch and it will take you ages ...

If you are a beginner like me, you should read the math chapters and all appendices of Carroll's book plus chapter 4. Then you should read a real book for beginners with a lot of examples how to apply GR in real calculations and how to understand it. For that I recommend James Hartle's "Gravity: An Introduction to Einstein's General Relativity" and Bernard Schutz's "A first course in General Relativity". After that hopefully you will understand the rest of Carroll's book better. My experience was that often I had to read Hartle's book in order to understand and solve a problem in Carroll's book.
... Read more

Book Description
Geometry is one of the oldest mathematical subjects in history. Unfortunately, few geometry study guides offer clear explanations, causing many people to get tripped up or lost when trying to solve a proof—even when they know the terms and concepts like the back of their hand. However, this problem can be fixed with practice and some strategies for slicing through all the mumbo-jumbo and getting right to the heart of the proof.

Geometry Workbook For Dummies ensures that practice makes perfect, especially when problems are presented without the stiff, formal style that you’d find in your math textbook. Written with a commonsense, street-smart approach, this guide gives you the step-by-step process to solve each proof, along with tips, shortcuts, and mnemonic devices to make sure the solutions stick. It also gives you plenty of room to work out your solutions, providing you with space to breathe and a clear head. This book provides you with the tools you need to solve all types of geometry problems, including:

• Congruent triangles
• Finding the area, angle, and size of quadrilaterals
• Angle-arc theorems and formulas
• Touching radii and tangents
• Connecting radii and chords
• Parallel, perpendicular, and intersecting lines and planes
• Slope, distance, and midpoint formulas
• Line and circle equations
• Handling rotations, reflections, and other transformations

Packed with tons of strategies for solving proofs and a review of key concepts, Geometry Workbook For Dummies is the ultimate study aid for students, parents, and anyone with an interest in the field. ... Read more

Customer Reviews (2)

For confused students
I recomend purchasing and going through the geometry for dummies first. This will only confuse you more if you don't already have a basic understanding. Once the basics are down, this is great practice.

Excellent Book!
Simple and easy to follow - a great book to work through if you need to learn proofs.Excellent book to help you prepare for the CSET single subject mathematics exam! ... Read more

Book Description

Building on the success of its first three editions, the Fourth Edition of this market-leading text covers the important principles and real-world applications of plane geometry, with additional chapters on solid geometry, analytic geometry, and an introduction to trigonometry. Strongly influenced by both NCTM and AMATYC standards, the text takes an inductive approach that includes integrated activities and tools to promote hands-on application and discovery.

• New! Tables provide visual connections between figures and concepts and help students better assess their level of mastery and test readiness.
• New! Chapter Tests have been added to the end of every chapter.
• New! Proofs have been varied to include written and visual proofs, as well as comparisons, to support students with different learning styles.
• New! Exercise sets in the Student Study Guide, with cross-references to the text, offer additional practice and review.
• New! Technology-related margin features encourage the use of the Geometer's Sketchpad, graphing calculators, and further explorations.
• New! Coverage now includes Section 2.6, Symmetry and Transformations.
• New! Technology Package includes the HM ClassPrep CD with computerized test bank (powered by Diploma).
• Updated! The number of Exercises and Explorations has been increased.
• Highly visual approach begins with the presentation of an idea, followed by the examination and development of a theory, verification of the theory through deduction, and finally, application of the principles to the real world.
• Discovery features reinforce the text's inductive approach: activities integrated throughout enable students to discover geometry concepts on their own, and section tools provide with hands-on application of geometric concepts
• Applications reinforce the connection of geometry to the real world: high-interest Chapter Openers introduce the principal notion of the chapter and relate to the real world and A Perspective On... sections conclude each chapter, providing sketches that are interesting, sometimes historical, and always informative.
• Summaries of constructions, postulates, and theorems are provided, and an easy-to-navigate numbering system for postulates and theorems provides a user-friendly structure. In response to user feedback, paragraph proofs feature more prominently in this edition.
• Comprehensive appendices include Algebra Review and An Introduction to Logic. A glossary of terms, a summary of applications in the text, and selected answers are also provided in the back of the text.

Customer Reviews (3)

Obscure and VagueThis Book Does It All
The instructions in this book are vague at best. The authors appear to be rushing the student through the fundamentals of Geometry without ever explaining how things work. The homework sections are a JOKE! The authors have given "solutions" to selct problems in the book, but do not explain how those answers are achieved. The book gives basics and ideas in the text then provides exercises with answers to some of the exercises. It would be much more helpful if they chose some of the problems to show a solution. Telling the student that the angle is 35 degrees does absolutely no good if the book does not tell the student WHY it is 35 degrees. If you have a great teacher who is willing to spend some time explaining in detail and going over the exercises in detail you may be o.k. with this book. If your teacher does not do these things then this book will hurt you more than it helps. If I could give it no stars I would have.

Worst Text Book EVER
Who am I to say this?I'm a 35-year-old college student with a 3.5 GPA, studying for a B.A. in English and Graphic Design in the Chicagoland area.I'm not that great in math but I with studying, good teachers and decent text books, I've passed all my math classes so far with As and Bs.I'm not the smartest person in the world but I am no dummy.Elementary Plane Geometry was required but I thought it might be interesting and maybe fun as I had NEVER taken a Geometry class before.

This book was completely useless.In fact, I ONLY used it to do the problems assigned to me by the teacher.I did not use this book to study nor as a reference guide.I used it as a coaster and a door stop a few times.My fellow students HATED it as well.

*There were NO examples of how to do the homework assignments.

*This book did not give you steps on how to solve problems.

*This book did not explain why an answer was what it was.

Opening up this book, to me, was like telling a 5-year old to go build a rocket and speak Chinese.

Instead of using this book, I barely passed the class by going every week to the tutors at school's study center (who, by the way, did not understand the "instructions" in this book either and had to read the chapters all the way through so they could teach me how the authors did the problems), my husband who is a mechanical engineer (who also hated this book and thought it was the worst math book he's ever seen) struggled through the text book with me, *and* I checked Geometry for Dummies out of the local library which helped a little.

If you are a school administration looking into using this book, I beg you to please not.

If you are a student that has been assigned this book, you should start lining up additional help now.

A Wretched and Horrid Book
Entering geometry for the first time ever can be an almost initimidating venture, especially for the first time geometry student.After conquering other mathematical courses successfully (like algebra and statistics), I presumed entering "Elementary Geometry for College Students," based on being given exceptional required texts from previous courses, would be a breeze.Oh how wrong I was!

"Elementary Geometry for College Students," by Daniel C. Alexander (of Parkland College) and Geralyn M. Koeberlein (of Mahomet-Seymour High School), has taught me nothing more than NOT to trust a textbook written by a couple of hack authors from unknown schools with a blatant disregard for meticulously explaining important vital and "elementary" steps as to how to arrive to certain statements, reasoning, deducing, measurements, and so much more NEEDED in successfully acheving full reign over geometry.For example, the origins of postulates and proofing are never explored, but slammed in your face, convoluting both topics along with breaking down statements from deducing a particular shape, its angles and measurements.The book's attempts at explaining triangles, convex polygons, congruent triangles, and properties of parallelograms are all but slandered together (with steps in basic algebraic mathematical equations arrogantly skipped over and presumed upon to you) without any form of thorough reason or explained steps bothered in explaining.Important theorems are disarrayed throughout with quick-step problem examples without helpful or detailed reasoning as to how the answer was ever achieved.

The authors have obviously assumed a college student has had some form of pre-geometry course prepping, and expect both instructor and student to know the advanced fundamentals without considering the beginner geometry student at all (just from judging by example and "solutions" given in each section).As a result, students will fail miserably, along with angering frustration, and discontent wonderment over what purpose geometry may ever serve toward a real-life career.Perhaps trying "Geometry for Dummies" by the infamous IDG publishing company would be a much suitable levelage to this otherwise detrimental book attempting to teach an important equation to the universe of mathematics.

By far, this textbook is the worst and most horrible book in teaching the subject of geometry!!

To professors searching a geometry book for your students: PLEASE avoid this book at all costs!You and your students don't need a textbook that presumes you know it all before diving into shapes, proofing, deducing, theorums, solids, and so forth.Most surely, there are much more superior books to this wretched and horrid title worthy of its decommissioning. ... 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
A flexible program with the solid content students need

Glencoe Geometry is the leading geometry program on the market. Algebra and applications are embedded throughout the program and an introduction to geometry proofs begins in Chapter 2. ... Read more

Customer Reviews (6)

The Epitome of What is Wrong in Math Education
Math books in Japan and Singapore are short, inexpensive, and to the point. Math books in American often look like this monstrocity: long, expensive and rambling. For example we learn about the history of jeans, we learn that Mae Jemison was the first African-American woman to go into space, and that in 1974, Beverly Johnson, the first black model on the cover of a major fashion magazine. In other words, this book tries very hard to hide the fact that is math book by introducing glossy photos, color pictures, and a litany facts TOTALLY UNRELATED TO MATH. Unfortunately, the students will eventually find out this is a math textbook, so introducing all of that fluff is a wasted effort.

In looking through this book, I would say about 65% of it can be jettisoned, leaving 35% of the book that actually deals with math in a direct and useful way. However, you really have to wade through a lot of nonsense to find that stuff.

Avoid this book. Period.

Perhaps the worst Math Book ever "Written"
In my home state of Tennessee, this is the text that unsuspecting school kids must decipher.It should have been easy - it's structure (and I use the term loosely) broadly resembles MTV with its numerous subjects per page, chats about unrelated subjects, some of the worst use of the English language ever and a colorful, multi-font, in-your-face appearance.There's pictures and graphs and arrows and charts and big text and small text and cartoons...Yet it cannot be understood much less absorbed - at least by anyone looking for something rational.

Like others, I noticed that the "team" of writers (it's almost as if they took turns writing paragraphs) continually introduced material BEFORE it was studied.Then there were the "examples" - just pitiful.Proofs were confusing, redundancy is taken for granted and the number of sub-subjects - review, standardized tests, chapter study, real life example, etc all ran together in a mushy mixture of words and concepts.IF YOUR CHILD MUST USE THIS BOOK GET EITHER A TUTOR OR PURCHASE "Geometry the Easy Way" by Lawrence S. Leff.I picked it up for $2.99 from Amazon.
A one star is overly generous.

Absolutely Horrid! Avoid like the plague!
I am taking an honors level geometry course at my high school, and this, unfortunately, is the text that we have been given. It's absolutely rubbish. The lessons don't really teach the concepts, they just show some examples, which are hard to comprehend without actually knowing the topic that they're trying to teach/demonstrate/show-off/whatever. Furthermore, the problems at the end of the lessons/chapters clearly assume knowledge beyond the scope of what was taught in the book (leaving holes for the teacher to fill in, I suppose). This book is bad, really bad, and I cannot stress this point enough.

I'd reccomend "Geometry the Easy Way" by Lawrence S. Leff, if you truly want to learn the topic (it covers everything that this joke of a text covers...only in more depth). Oh, and it's only $10 or so.

Also, avoid the Algebra II textbook by these clowns, it's written in the same bad style, poorly teaching the lessons. In general, Glencoe products are of the lowest quality. I'm studying calculus independently, and I'm sure glad that Glencoe doesn't produce a calculus text (otherwise the school may have given me that).

Utterly Confusing for Middleschoolers
I was recently amazed when my daughter, who has been a strait A math student, showed me this textbook along with a poor grade in her first quarter taught with this book.I am an electrical engineer with math minor who greatly enjoys math applications in engineering.

I've read through many chapters of this book in utter confusion.There are so many applications and examples, it's hard to find the math concepts, particularly in the early introductory chapters.Chapter one of this book, used to teach 8th graders in our school district, reads like a textbook on vector calculus.The discussion changes between 1,2, and 3 space without any discussion or any definitions.Almost any 1960's geometry textbook would be far superior.

In reading later chapters, it seems to cover boolean logic, logic theory, matrices, then returns to geometry.The trigonometry sections are fairly good.Towards the end, a more standard approach is taken towards solid geometry.The homeworks are very confusing, and relate to materials not in the chapters.

This book has just toomuch confusing stuff for middleschoolers.I'm sure that it's well intended, but completely misses the mark.It fails to teach geometry.Chock full of applications, especially in the early part of the book, it's missing the fundamental ideas of geometry. It's also very weak on analytical geometry, the most important ideas in engineering.

This textbook was a poor choice in our school district.The latter chapters are almost college level in complexity.I'm sure that the egos of the authors were well massaged in coauthoring this.However, the students are left without a useful textbook that they can understand or learn math from.If making students thoroughly confused and feeling inadeqate is a new concept in education, then this book is great.No wonder students can't pass the Aims test.

Geometry: Misconnections
This is a very poor math textbook.Here's the main problem: End-of-chapter problem sets require knowledge and skills that have not been clearly introduced in the chapter.Chapters meander through irrelevant or mundane points (e.g., how to bisect an angle using a compass...this is middle school stuff) but then call on different skills in problem sets (e.g., where was this postulate discussed in the chapter?).If you are looking for a good geometry book, I suggest that you look elsewhere.Unfortunately, our school district didn't. ... Read more

Book Description
This text has been adopted at: University of Pennsylvania, Philadelphia University of Connecticut, Storrs Duke University, Durham, NC California Institute of Technology, Pasadena University of Washington, Seattle Swarthmore College, Swarthmore, PAUniversity of Chicago, IL University of Michigan, Ann Arbor "In the reviewer's opinion, this is a superb book which makes learning a real pleasure."ne de Mathematiques Pures et Appliquees"This main-stream presentation of differential geometry serves well for a course on Riemannian geometry, and it is complemented by many annotated exercises."F. Mathematik "This is one of the best (if even not just the best) book for those who want to get a good, smooth and quick, but yet thorough introduction to modern Riemannian geometry."es MathematicaeContents: Differential Manifolds * Riemannian Metrics * Affine Connections; Riemannian Connections * Geodesics; Convex Neighborhoods * Curvature * Jacobi Fields * Isometric Immersions * Complete Manifolds; Hopf-Rinow and Hadamard Theorems * Spaces of Constant Curvature * Variations of Energy * The Rauch Comparison Theorem * The Morse Index Theorem * The Fundamental Group of Manifolds of Negative Curvature * The Sphere Theorem * IndexSeries: Mathematics: Theory and Applications ... Read more

Customer Reviews (8)

Needs a table of symbols
This is another well-written text by Do Carmo. I browsed through it and found I could not understand several passages because I did not know what the special symbols meant and there was no table of symbols. I plead with the publisher to add such a table to the next edition or printing.

The best introduction to Riemannian Geometry I have read
Most books about Riemannian Geometry can be quite difficult to read and understand if you are not lucky to have a good teacher that explains you the contents in a more intuitive way. This is not the case of the present book. Of course good teachers will not do any harm, but I would dare to say that this book can be read through and understood without their help. The exposition and the proofs are elegant and concise, but one still gets all the essential intuitive hints to grasp the fundamental notions in RG.

This has been my experience with RG: I did not have good teachers (of RG), and the first 4 or 5 books I tried to study discouraged me and made me think I could never understand RG. Until I found Do Carmo's book. Then I found out that RG was intelligible.

I consider this book the best available introducion to RG in less than 300 pages, and I would definitely use it if I had to teach RG. Of course some important topics must be missing in a relatively short introduction, and if one wants to continue studying RG other books (such as Sakai's one) should be studied as well, but for quickly acquiring a sound basis of RG this is the best I know of.

Concise and clear
This is really a very good book to start Riemannian Geometry (RG). Exposition of key concepts of RG (affine connection, riemannian connection,geodesics, parallelism and sectional curvature, ...) are well motivated and concisely explained with numerous motivating and not so difficult execises. The book is self contained convenient for self study. It contains an introductory chapter on mathematical background explaining basic concepts as differentiable manifolds, immersion, embedding and so on, which are necessary to deal with RG. I have essentially one basic remark about this book. Formulation of RG as presented in it, is a little bit dated. Now, with the development of geometric algebra and Geometric calculus most, if not all, mathematical concepts needed to study RG like covariant derivative, curvature, and general tensors can be formulated without ressort to coordinates and in a manner to highlight their essential geometric features. Moreover derivation of certain formulae can be much easier and natural. For example the author defines the formula for |x^y| as sqrt(sqr(|x|).sqr(|y|)-sqr(inner product(x,y))). Then explains that it is the area of two dimensional parallelogram determined by the pair of vectors x and y. The reader might be puzzled as to how this formula is obtained. In the context of geometric algebra this is derived very naturally from basic concepts. Anyway, this remark does not diminish the value of this book.

Best 1st semester Riemannian Geometry book after 1 semester DG
This is the best Riemannian Geometry book after students have finished a semester of differential geometry.It gives geometric intuition, has plenty of exercises and
is excellent preparation for more advanced books like Cheeger-Ebin.

Students should already know differential geometry (Spivak "Calculus on manifolds" and Spivak "Differential Geometry Volume I" might be used there)

Warning: the curvature tensor is defined backwards as compared to Cheeger-Ebin.

Definitely a good start
This book is definitely a solid way to start in Riemannian geometry. The topics chosen give a glimpse of more advanced topics that the reader can venture to next, and the order covered leaves little confusion. The book is to the point, with little conversation about the concepts except at the very beginning of each chapter.

I only have two complaints, but neither would cause me to lower the rating to 4 stars.

1. There could be more "deep" exercises that allow the reader to explore more of the subtleties of the subject. And for what exercises there are, the author sometimes gives far too much away in "hints."

2. The book does not take a unified approach to the subject that fits nicely with the full generality of the theory. This is probably what makes the book good to start with, but there is still going to be a somewhat difficult transition from this book to a typical differential/riemannian geometry book. Namely, the basic language of vector bundles, pull backs/push forwards, tensors and tensor fields are either covered in a very specific framework or not at all. ... Read more

Book Description
An Unabridged Printing, To Include All Figures: Lemoine And Brocard Points - Angular And Tripolar Coordinates - Pedal And Antipedal Triangles - The Medial Triangle - Simson's Line - The Orthopole - Orthogonal Projection ... Read more

Book Description
A good knowledge of geometry is essential to success on many standardized tests and applicable to a wide range of careers. Geometry Success in 20 Minutes a Day provides a thorough course in geometry skills that can be fit into any busy schedule. ... Read more

Customer Reviews (1)

Geometry-success in 20 minutes a day
I am a homeschooling mother of 2 high school students.I have been teaching geometry but with great difficulty.It has been 30 years since I have taken geometry and it was not a happy experience then and a strugglenow.We do use a regular textbook but they are not always as clear a Iwould like them to be.I have been using this book as a back up inteaching the basics.They are clearly presented and easy to grasp.Eachchapter presents a concept with self tests to make sure you understandbefore you go on the the next idea.It is a great suppliment to ourtextbook and a good review for the GED test. ... Read more

Customer Reviews (2)

This is book is completely lacking in everything.
This book is lacking in examples, diagrams and as well as facts and information. The diagrams that are present do not provide the necessary information to actually LEARN form the book. As a result, we have to deviate from the textbook making it completely comfusing. DO NOT BUY THIS BOOK. It is simply not worth it.

I suggest Prentice Hall if you need a book, after using Prentice Hall for several years; I find it completely helpful and easy to learn from.

The book has few examples and it doesn't show HOW to do the problem, rather, it just explains what the problem is and quickly gives the solution. The chapters are organized in a fashionable manner but it is lacking in material, the answers are at the back of the book but what really ticks me off is the 'real life problems'. They give 4-5 of those in a lesson and they take at least half an hour of think and writing [that doesn't involve math' and then on the test, they never show up becuase the problem had nothing to do with the chapter. The book is just lacking in materials. Look at the publiciation date. 1998--this book is just wayyy out dated.

The perfect one!
Now this is type of Geometry book it should be. More examples, answer key, and a glossary. Unlike that complicated Michael Serra Geometry book from hell. ... Read more

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 book is intended as a textbook for a first-year graduate course on algebraic topology, with as strong flavoring in smooth manifold theory. Starting with general topology, it discusses differentiable manifolds, cohomology, products and duality, the fundamental group, homology theory, and homotopy theory. It covers most of the topics all topologists will want students to see, including surfaces, Lie groups and fibre bundle theory. With a thoroughly modern point of view, it is the first truly new textbook in topology since Spanier, almost 25 years ago. Although the book is comprehensive, there is no attempt made to present the material in excessive generality, except where generality improves the efficiency and clarity of the presentation. ... Read more

Customer Reviews (6)

a different perspective
While I agree with reviewers generally that this is a good book, i should warn that bredon isnt for the faint of heart. He makes use of simple language from category theory, doesnt always completely introduce his discussions (see for example the chapter on the tangent bundle where tangent bundle is never defined), and some other things that are nuisances to the newcomer.

I do think this is a good modern readable textbook, but for the student who has a solid foundation in mathematics. I didnt find it as accessable as other topology books, say Hatcher or Lee's books (but lee's are not as complete).

excellent for first year graduate study
This was the assigned book in my first year grad topology course. It has good examples, interesting exercises. I like the emphasis on geometrical examples, constructions. It's not easy to read, but interesting.

Among the best textbooks in algebraic topology.
As the previous reviewers have commented, this book is very accessible and detailed. I should add that the authour never lets you get lost in the labyrinth of abstract nonsense; the treatment is always geometric rather than homologico-algebraic. The only complaint I have is, the book would be more useful with chapters on spectral sequences, cobordism and K-theory.

The Graduate Sudent's Topology Bible
If you want to learn topology, this book is the place.Though this text can require some maturity, the range of topics and the clarity of exposition are outstanding.My only complaint is that an additional appendix covering the basics of category theory would have been quite useful.Bredon not infrequently uses the language of category theory (though always in a non-essential way).Since this text is aimed at 1st year graduate students, I think the tacit assumption that the student has already encountered these topics is not justified.That such a minor point is my chief complaint speaks volumes of my esteem for this text.

My secret weapon in topology
Today I told someone that Bredon's "Topology and Geometry" book was my secret weapon.I say this because it has mostly everything a grad student in topology needs to know in order to be fluent in the subject.Ibought it not because I was taking a class from it--I got it because I sawit at the library and realized that this is the 'end all' topology book forme and that I will not need to buy all those 'other' books in topologyanymore.I will warn you though--some of the proofs are terse, so you haveto be somewhat 'mathematically mature'.It is definately harder thanMunkres' "Topology: A first course" (the red book).But it goesbeyond the point set crap that gets (in my opinion) too much airtime.Iwish I would have taken a class under this book. ... Read more

Book Description
An ideal program for struggling students

Geometry: Concepts and Applications covers all geometry concepts using an informal approach. ... Read more

Customer Reviews (2)

Great examples and relatable!
Having taught from this book, I think it's a great curriculum to use with students who aren't interested in theory, but would rather know "why are we doing this?"This book branches out into how geometry is found in culture, such as art, and applied in daily living.If you are more interested in answering students' questions about "when are we ever going to use this?" than you are about writing proofs all the time, this is a perfect book for you.

this is the worst
I love mathematics and read math book of all sorts, including textbooks. I teach music theory (which is similar to math) at the college level, and have been doing so for the last 18 years. This is the worst math book I have ever seen. Where are the brillant proofs of Euclid? This is a hodgepodge of history, so-called music, and diverse topics thrown into one book, that consists of mearly formulaes and prove it yourself theorems. The Pythagorean formula is circular, if you can follow it back far enough. There are no concepts or brillant ideas, only "you do it this way" as in the old calculus books. The authors are all school teachers, with one college teacher (wow). The school teachers have been complaining for years about how bad the "new math" was when the mathematicians wrote the texts; now it is their turn and they have turned my child off to math by their dull pedantic book. I think all schools should review this book before using it. ... Read more

Book Description
Basic Algebraic Geometry, Volume I, is a revised and expanded new edition of the first four chapters of Shafarevich's well-known introductory book on algebraic geometry. The author has added plenty of new, mostly concrete geometrical material such as Grassmannian varieties, plane cubic curves, the cubic surface, degenerations of quadrics and elliptic curves, the Bertini theorems, normal surface singularities. There are also some new number-theoretical applications. Shafarevich succeeds in making algebraic geometry accessible to non-specialists and beginners and his two-volume book will remain one of the most popular introductions to this field. The book is suitable for third-year undergraduates in mathematics and also for students of theoretical physics. ... Read more

Customer Reviews (2)

Someone hasn't read the first page of the index!!!!!!
I have been a student of AG for the past six years and I have come to the conclusion that Shafarevich is a great place to start.Having said this, one must have the necessary background in algebra and topology.I disagree with the other reviewer about doing this after Hartshorne--start here then do Hartshorne!!!Oh ya, the index refers to both volumes 1 and 2; read the first page of the index!!!

After Hartshorne!!!
This book is very good for the secondary course after learning with Harshorne's Algebraic geometry. ... Read more

Book Description

Customer Reviews (1)

A Classic on Euclidean geometry
Recently Dover has reissued two classics on Euclidean geometry, College Geometry: An Introduction to the Modern Geometry of the Triangle and the Circle (Dover Books on Mathematics)and this book by Johnson. Both books were originally issued in the first half of the 20th century and both were aimed at a college level audience. Both of them also have a considerable amount of so called triangle geometry. As triangle geometry has seen a large upsurge the last years there is certainly a need for an English book that gives an overview of the subject. These books are useful in this respect but are out of date. Until a modern treatment of the subject (The Triangle Book by Conway and Sigur for instance, but when when ... ?) will be available, these two books and the resources on the www will have to do. Altshiller Courts' book has a great set of exercises that can be used as a training ground for geometric problem solving. The problems in Johnsons' book mostly ask for proofs of theorems that are ommited in the text (that's why I give 4 stars). If you are interested in the subject, buy both, its certainly value for money.

The book assumes that you are familiar with simple geometrical concepts like congruence of triangles, parallelograms, circles and the most elementary theorems and constructions as can be found in Kiselev's book Kiselev's Geometry / Book I. Planimetry. ... 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
This is a reissue of Professor Coxeter's classic text on non-Euclidean geometry. It begins with a historical introductory chapter, and then devotes three chapters to surveying real projective geometry, and three to elliptic geometry. After this the Euclidean and hyperbolic geometries are built up axiomatically as special cases of a more general 'descriptive geometry'. This is essential reading for anybody with an interest in geometry. ... Read more

Customer Reviews (1)

The beauty of geometry is captured
Originally published in 1942, this book has lost none of its power in the last half century.It is a commentary on the recent demise of geometry in many curricula that 33 yearselapsed between the publication of the fifthand sixth editions. Fortunately, like so manythings in the world, trendsin mathematics are cyclic, and one can hope that the geometriccycle is onthe rise. We in mathematics owe so much to geometry. It is generally conceded that much of the origins of mathematics is due to the simplenecessity ofmaintaining accurate plots in settlements. The only book fromthe ancient history of mathematics that all mathematicians have heard of isthe Elements by Euclid. It is one of the most read books of all time,arguably the only book without a religious theme still inwidespread useover 2000 years after the publication of the first edition. The geometry taught in high schools today is with only minor modifications found in theEuclidean classic.
There are other reasons why geometry should occupya special place in our hearts. Most of the principles ofthe axiomaticmethod, the concept of the theorem and many of the techniques used inproofs were born and nurtured in the cradle of geometry. For manycenturies, it was nearly anact of faith that all of geometry wasEuclidean. That annoying fifth postulate seemed so out ofplace and yet itcould not be made to go away. Many tried to remove it, but finally theHolmseandictum of ,"once you have eliminated the impossible, what isleft, not matter how improbable,must be true", had to be admitted. Therewere in fact three geometries, all of which are of equalvalidity. Theother two, elliptic and hyperbolic, are the main topics of this wonderfulbook.
Coxeter is arguably the best geometer of this century but therecan be no argument that he is thebest explainer of geometry of thiscentury. While fifty years is a mere spasm compared to thetime sinceEuclid, it is certainly possible that students will be reading Coxeter farinto the futurewith the same appreciation that we have when wereadEuclid. His explanations of thenon-Euclidean geometries is so clear thatone cannot help but absorb the essentials. In so manyways, Euclideangeometry is but the middle way between the two other geometries. A pointwellmade and in great detail by Coxeter.
Geometry is a jewel thatwas born on the banks of the Nile river and we should treasure and respectit as the seed from which so much of our basic reasoning processessprouted. For thisreason, you should buy this book and keep a copy onyour shelf.

Published in Smarandache Notions Journal, reprintedwith permission. ... Read more

Book Description

Customer Reviews (19)

Instructive, and interesting
I found the book to be both educational, in that I learned great deal about geomtery and the history of diemsions from this book, as well as being fun to read. Both interesting and intellectually stimulating--I find this combination rare. I recommend ths book to anyone interested in the field.

With few exceptions, it is a readable, stepwise explanation of how the universe is structured
To understand relativity, it is necessary to understand geometry, specifically how a straight line can be curved. For nearly everyone, any attempt to understand four-dimensional space begins with understanding how a three-dimensional creature would appear to a two-dimensional one. One of the earliest and still the greatest of all introductions to going up a dimension is "Flatland" by Edwin A. Abbott. Quite naturally and sensibly, Rucker starts with Abbott's rendition of the properties of Flatland.
Rucker then moves on to the idea of curved space, where the shortest distance between two points is a "straight line", which is curved by the properties of the space. The space that we occupy is curved by the presence of matter, as Einstein claimed in his relativity theories. Furthermore, movement causes shrinkage in the direction of the movement and the slowing of time, which causes time to become just another dimension of space. As counterintuitive as this may appear, Einstein's relativity theory has been verified over and over again to a large number of significant figures.
One of the best things about this book is that Rucker has included problems at the end of each chapter. These problems reinforce the concepts of the chapter; it is unfortunate that no solutions were included.
In this book, Rucker steps the reader through all of the background material necessary to understand relativity and four-dimensional space. With few exceptions, the accounts are understandable to anyone with an understanding of college algebra.

The best book ever in its field
This book has presented the most difficult topics of our world with the easiest words. After reading this book many of my questions that I had in my mind for a long time were answered. It's worth thousands more than its price.
Congratulation to Mr. Rudolf Rucker for his great book.

explain dimensions very well
it is publishedyears beforebut it is almost new for today and it explain dimensions and shape of space well and clearly .thanx to amazon for sending me timely.

See what's outside the box
Over two millenia ago, Euclid wrote his masterpiece Elements and stated in his fifth postulate that only one perpendicular line could pass through any one point adjacent to another line.

One hundred fifty years ago, it was proven that yet another geometry could be described by asserting that more than one parallel line could pass through such a point.

Building on these ideas, Rucker briefly yet thoroughly surveys the relevant mathematics outside the box of Euclidian geometry.

It's a fascinating place too because it involves considerations of hyperspace, four dimensional travels and ultimately Einstein's theory of relativity.

Copiously filled with illustrations to help drive home his points, Rucker has produced a book that meaningful helps one visualize and better understand the fourth dimension.

This book is an excellent read along with Choas, Coincidences and All that Math Jazz, The Fourth Dimension Simply Explained, Einstein's own Relativity and Hyperspace by Michio Kaku which discusses all these ideas as well as contemporary string theory (which purports to pull it all together). ... Read more

Book Description
Among the many beautiful and nontrivial theorems in geometry found in Geometry Revisited are the theorems of Ceva, Menelaus, Pappus, Desargues, Pascal, and Brianchon. A nice proof is given of Morley's remarkable theorem on angle trisectors. The transformational point of view is emphasized: reflections, rotations, translations, similarities, inversions, and affine and projective transformations. Many fascinating properties of circles, triangles, quadrilaterals, and conics are developed. ... Read more

Customer Reviews (5)

Great book
This is a wonderful book if you want to gain a real understanding of what geometry can be (if you like this book you should buy the biography of Coxeter: "King of Infinite Space: Donald Coxeter, the Man Who Saved Geometry" - it is one of the best biographies of a mathematician on the market and shows that Coxeter was a genius and a hoot).However, if you know a significant amount of geometry, then try "Introduction to Geometry, 2nd Edition" as that is the more complete (and (very) much more rigorous) text.Also, if you like this book, then buy "Visual Complex Analysis"
by Tristan Needham - it is a great companion to Brown & Churchill.

Incredibly helpful
I just worked through this book & I have to say that it's probably the best for "what's next" after high school geometry. It's got a chapter on each of the following things: triangles (with Ceva's & Steiner-Lehmus' theorems, orthic triangles & Euler lines, etc), circles (like power of a point, etc), collinearity & concurrence, transformations (Yaglom's series of 3 books are better for these though but this is a good introduction), and an intro to inversive geometry and projective geometry. One section that blew my mind is the one about the "3-jugs" problem where you've got a jug that holds 8L, one that holds 5L & one the holds 3L and the goal is to get 4L in each of two jugs. Maybe you've heard of it...? This book gives a general method for solving problems like that using reflections. I liked how each section is very short and typically has just 2-3 theorems, so the book is split into manageable pieces. Maybe it's just a psychological thing, but I like that. Depending on the section, there are 3-10 problems at the end of each section; like any book there are easy ones with fairly obvious solutions and there are hard ounes that take a lot of thought. I think the hard ones might have come from olympiads because Sam Greitzer was a USAMO coach at some point. It doesn't matter though, this book gives you a lot of tools you can use to deal with olympiad-level problems. A couple good ones to work on after this book would be Johnson's "Advanced Euclidean Geometry" and Court's "College Geometry", both out of print unfortunately but maybe a nearby library wold have them?

THE BEST SOURCE FOR GEOMETRY
I am a high school student who competes in extracurricular mathematics contests. I've found many great books for trigonometry, algebra, precalculus, calculus, and any other field of math except geometry, i.e. until I found this book. A thorough knowledge of geometry beyond what you learn in high school in necessary needed to be competitive, and the place to get it is Geometry Revisited. This book covers any geometry you could need and is by far the best, if not the only, book out there with what you need to know. This book was also very interesting and thought provoking. The selection and arrangement of topics couldn't be any better. Each lesson even contains exercises (with answers and hints) to show you how to apply what the book teaches you. I would recommend this book to any high school student competing in any contests, to any person interested in extending their knowledge of geometry, or to anyone who with any interest in math. I can tell you that you will learn multitudes and enjoy the book too.

FOR ALL THOSE INTERESTED IN MATHEMATICS
This is 'the book' the typical high school student in America needs to read in order to see what Geometry is really all about.Geometry can be fun and engaging. Unfortunately it has taken a back seat in the high school curriculum.This is unfortunate considering the beauty that the subject has to offer.Many topics are covered in this book:The extended law of cosines, Ptolemy's theorem, transformational geometry, conics, and many, many, many more. This is also a book that is essential for mathematical problem solvers-- particularly for those interested in competing.All that is needed to follow along is the basic geometry learned at the high school level.This book is a must have for any mathematical library.

A++ Highly recommended.

The Best Undergrad Level Geometry Book Ever
Geometry Revisited should be used as a textbook for every geometry classin high school. It offers comprehensive proofs and theorems that everyundergrad should know by the time he/she gets through basic geometry. Thelast sections of this book also offers a slight touch on comcepts such asnon-Eucledian and projective geometry, serving as a launchpad to higherlevels of education. I highly recommend this book to any high school orundergrad teachers and students. ... Read more