Product Description
This book is an introductory graduate-level textbook on the theory of smooth manifolds. Its goal is to familiarize students with the tools they will need in order to use manifolds in mathematical or scientific research--- smooth structures, tangent vectors and covectors, vector bundles, immersed and embedded submanifolds, tensors, differential forms, de Rham cohomology, vector fields, flows, foliations, Lie derivatives, Lie groups, Lie algebras, and more. The approach is as concrete as possible, with pictures and intuitive discussions of how one should think geometrically about the abstract concepts, while making full use of the powerful tools that modern mathematics has to offer. Along the way, the book introduces students to some of the most important examples of geometric structures that manifolds can carry, such as Riemannian metrics, symplectic structures, and foliations. The book is aimed at students who already have a solid acquaintance with general topology, the fundamental group, and covering spaces, as well as basic undergraduate linear algebra and real analysis. John M. Lee is Professor of Mathematics at the University of Washington in Seattle, where he regularly teaches graduate courses on the topology and geometry of manifolds. He was the recipient of the American Mathematical Society's Centennial Research Fellowship and he is the author of two previous Springer books, Introduction to Topological Manifolds (2000) and Riemannian Manifolds: An Introduction to Curvature (1997). ... Read more

Customer Reviews (19)

My key to Differential Geometry: A subjective review
I have in mind the following reader:
someone who will do what it takes to learn Differential Geometry.

The title of this book is not 'Differential Geometry,' but 'Introduction to Smooth Manifolds;' a title I think is very appropriate. In this book, you will learn all the essential tools of smooth manifolds but it stops short of embarking in a bona fide study of Differential Geometry; which is the study of manifolds plus some extra structure (be it Riemannian metric, Group or Symplectic structure, etc).I should note, however, that it does cover elementary notions of Riemannian metrics and a fair amount of Lie Groups. At first I found it annoying that I had to work through over 500 pages of dense mathematics before I could study what I really had my heart set on: Riemannian Geometry. But, having read Lee's book cover to cover, I am glad that I waited and developed all the necessary tools.

Lee assumes the reader is well prepared, i.e. has had rigorous courses in Multivariable Analysis especially up to Inverse Function Theorem at the level of, say `baby Rudin' (but, Lee does prove this is complete detail), Group theory, Linear Algebra, and Topology. In my opinion, all of these are necessary for a deep understanding of the subject.

I would advice anyone who will work through Lee's tome to pick up a slimmer, more concise book to stay relatively grounded. My personal favorites are: Janich's `Vector Analysis' (I can't recommend this enough!), Barden's `Intro to Differentiable Manifolds,' Janich and Brocker's `Differential Topology' (hands down, the best pictures!), Milnor's `Topology from a Differentiable Viewpoint.'

** Merits **

-Pedagogical, motivational, student friendly (Excellent Index!), lots of details
-Moves slow, takes its time developing basics with lots of pictures and heuristic arguments
-Lots of worked out examples!
-Very good selection of problems
-Very useful appendix on Topology, Analysis, and Linear Algebra (A must read as the highlights of the subjects are conveyed with only the useful proofs thrown in)
-Prepares one for advanced books in Differential Geometry, i.e. Riemannian Geometry, Differential Topology, etc.
-The entire book can be covered in a semester and a half, leaving time to cover most of Lee's Riemannian geometry book.


** Simultaneous Merits, Stumbling blocks, and/or Distractions **

-Too much information for a first reading
-Too wordy (overly detailed in proofs)
-Subjects are introduced at the moment tools are available, not in their own separate chapters
-Not clear how chapters are interdependent (however, research mathematics is not artificially divided so it's refreshing to read a book that embraces this)

** Faults/Disadvantages **

-Lots of typos, so be sure to download the list of errata from the authors webpage
-Style not for everyone; some readers will prefer more reserved, concise treatments. To this end, I can recommend Warner's `Foundations of Differentiable Manifolds and Lie Groups'
-Need to look elsewhere for Riemannian geometry, i.e. there is no mention of a connection or curvature
-Not useful as a reference (unless, of course, you worked through it cover to cover and a have a feel for when things were introduced. However, the index is excellent)

Please have a look at the reviews by Mr. Raleigh and "math reader." I agree whole-heartedly with their assessment of Lee. They also talk about some aspects that I do not repeat =)

** Conclusion **

This book is incredibly addicting and FUN to read, work from, and learn!

Excellent Book!
Yes, this book is almost 550 pages (plus a long appendix reviewing topology, linear algebra, and analysis), but I find the pacing simply perfect.The extensive lists of examples are helpful and the exercises are challenging but doable.Yes, the printing quality could be better, but it does not affect the text, and the price can't be beat for such a readable and useful book.It is simply the best choice out there for self-study of manifold theory.

The bestbook I have read on the subject
I have read some books on this subject Spivak, Munkres , SS Chern, Warner and I think this is simply the best. Clear, motivating ...

Thanks professor Lee

Only one small detail: the tangent space consturction of SS chern (lectures on differential geometry) is perhaps a bit better but it`s rather difficult to appreciate it at first time.

Beautiful! Welcome to graduate school.
Simply, this book is fantastic! I am a graduate student in mathematics taking diff. top. and we are using this book. We have covered ch 1-7, 11-15 so far, and are now going back to finish immersion theory, with Milnor as a second reference. Most of this class I have taught myself the material directly out of Lee. The book is mind blowingly good for self study. I believe that an undergraduate with A. Calc and point set topology can work through this book and understand it well.

A few suggestions;

-worry less about the in-text exercises and more about the end-text exercises.
-go through the appendix before you begin, its easy and quick, but useful
-read with great detail. Lee provides everything, you may just not see it at first.

The printing is not up to the standard of the writing
By all accounts, this and Dr. Lee's other two books on manifolds are exceptionally well-written.But my copies arrived from Amazon this week, and, unfortunately, Amazon and Springer have decided to replace the crisp offset-printing of earlier printings by lower quality digitally-printed versions, probably as a cost-cutting measure.

If you care about how books look, I'd suggest trying Amazon marketplace or small retailers elsewhere to increase your odds of getting a superior copy from an earlier printing.
... Read more

Product Description
The year is 2020. Fueled by an insatiable curiosity, Reid Malenfant ventures to the far edge of the solar system, where he discovers a strange artifact left behind by an alien civilization: A gateway that functions as a kind of quantum transporter, allowing virtually instantaneous travel over the vast distances of interstellar space. What lies on the other side of the gateway?Malenfant decides to find out. Yet he will soon be faced with an impossible choice that will push him beyond terror, beyond sanity, beyond humanity itself. Meanwhile on Earth the Japanese scientist Nemoto fears her worst nightmares are coming true. Startling discoveries reveal that the Moon, Venus, even Mars once thrived with life, life that was snuffed out not just once but many times, in cycles of birth and destruction. And the next chilling cycle is set to begin again . . .
Amazon.com Review
Stephen Baxter follows up his Arthur C. Clarke Award nominee Manifold: Time with thesecond book in the Manifold series, Manifold: Space. In this novel,former shuttle pilot and astronaut Reid Malenfant meets his destiny onceagain in a tale that stretches the bounds of both space and time.

The year is 2020 and the Japanese have colonized the moon. The 60-year-old Malenfant is called there by a young scientist named Nemoto who has discovered something in the asteroid belt that can only mean humans arenot alone in the universe. The aliens seem robotic in nature and appear tobe building something in Earth's backyard. The Gaijin, as theyare called by humans, don't respond to communication efforts so an unmannedship is launched to investigate. In the meantime, Malenfant decides answersare only possible by mounting an expedition to Alpha Centauri, which may bewhere the Gaijin come from.

Baxter, who won the John W. Campbell Award and the Philip K. Dick Awardfor his novel The Time Ships, orchestrates a stunning array of scientific possibilities in Manifold: Space. Each chapter adds a new piece to his mosaic of humanity's future. The novel is admirable in its enormous scope, but it's hardto invest much emotion in the characters. Although they are well drawn,they vanish for long periods of time as Baxter leapfrogs through time and space.Manifold: Space, by its nature, lacks passion but excels in grand ideas. --Kathie Huddleston ... Read more

Customer Reviews (39)

Cruel, austere, but immensely enjoyable.
(If you're not here for a spoiler, don't read past the spoiler line!)

If you're looking for jarheads in space, egyptian aliens, pod races and feel good endings about humankind's unique humanity you can step away now.

The rest of you can face the bloody harsh wind of Baxter's second attempt at explaining the Fermi paradox: if life just happens, why have we only found it on our planet? The book takes a long, descriptive path through the next few thousand years, painting on a grand scale what's going on in our stellar neighborhood. Relatively convincing quasi physics are woven into a surreal picture of space/time travelers, incomprehensible aliens, resource struggles on a ridiculous scale, and ultimately a cold, cold, unfortunately teetering universe.

The payoff for sticking with this very deliberately paced revelation of the universe's mechanical heart is gaining a moment of clarity where you get to pose yourself the question: If the universe is "just" the universe, sans the metaphysical we've imagined, if it really is just a big cooling explosion and we're just bizarre knots in the eddies of cooling gasses... do waffles and bacon still taste good?


***** HERE BE SPOILERS *****

For those who'd like a few more details before committing to the book:
Malenfant is back, and ends up with a key role again. He isn't the same Malenfant, just as this isn't the same universe as the first Manifold book. The idea here seems to be that Baxter is using each book to explore another answer to the Fermi paradox. Some of the first book's characters make cameos here, but there's a new cast to make room for too.

This time around, the answer to the paradox revolves around the idea that life does happen all over the place, but it gets wiped out all the time. This seems to happen on a period that allows for a sufficiently high degree of sophistication that the relics of those civilizations are incomprehensible to us. Baxter would like us to picture the idea thatsufficiently advanced resource plundering is indistinguishable from geology. As with the first book, he also casually tosses star engineering out there as part of his mechanism. Given the nature of the relics, the timescales involved, and the thoroughness of the "reboot" function, he paints a convincing picture of why we aren't seeing any evidence of others in the sky.

As per the first book, we follow our ragged collection of human observers out to the stars to observe all of this first hand en route to the melancholy final reveal. The pacing is slow and Baxter spends what feels like more time describing every nut and bolt compared to the first book. Given that the whole point of the book is to paint you a picture of this tragic universe though, this really is only a bad thing if you have a short attention span.

The punchline, the bit about our part to play, is refreshing. No magic invoked here, no triumph of vague spirit or ill defined virtue. Baxter calls us out for being what we manifestly demonstrate ourselves to be rather than what we'd like to think we are, and finds a use for it!

I very much enjoyed this book. Two thumbs (flecked with maple syrup) up.

Brilliantly imaginative - a novel of epic proportions
"Space" is the second book in Stephen Baxter's Manifold trilogy, and a sequel of sorts to "Time", although it can also be read independently. Once again the central character is Reid Malenfant, an ex-NASA astronaut and failed entrepreneur. Obsessed with the search for extraterrestrial life, Malenfant seeks a solution to the Fermi paradox: given that the universe is billions of years old, if life exists out in the cosmos, why don't we see the evidence of it all about us? Thus when alien intelligence is detected out in the asteroid belt, Malenfant takes it upon himself to investigate, to make contact and ultimately to follow them back to the stars, through the mysterious blue portals through which they came.

The action unfolds over no less than 1,800 years, from the present day up to the thirty-eighth century, with the final, epic conclusion set another 5,000 years after that. In this way Baxter lays out a compelling vision of the possible long-term effects of Earth's contact with aliens. Unlike in "Time", where he employs an interesting mix of faux newspaper articles, blogs and journal entries to tell his story, in "Space" he sticks to a more conventional third-person narrative. The story is related through the perspective of four or five main characters, all of whom use the portals to travel to the stars and see life beyond Earth, and who, over the course of many years, become witnesses to the gradual decline of human civilisation.

The story is episodic in nature, and has the impression of a number of short stories loosely linked together. This can be frustrating for the reader, as there are enough intriguing ideas packed in this book to sustain half a dozen different novels. Each successive world is imaginatively drawn - from Earth, Io, Triton and Mercury to Alpha Centauri and far beyond - but Baxter tends to pass over them all very quickly, which does become tiresome. There comes a point about two-thirds of the way in when one wonders what the ultimate point is. Another result of the disjointed nature of the novel is that is difficult to feel fully engaged with the characters or get a sense of their development in these extraordinary circumstances. It is disappointing, too, that Malenfant - in principle a fascinating character - does not feature more, despite his centrality to the story. However, it is clear that this is not meant to be a character-driven novel so much as one based around ideas. Indeed "Space" has at its heart themes of human ambition and determination, consciousness and identity, self and soul, and the will to survive in a hostile universe, all of which are explored in depth.

In "Space", the author shows an imagination and consideration of the big questions of existence which is not often seen in most modern SF. It is true that there is less hard science and more scientifically-informed speculation than there was in "Time", but Baxter delivers it with such confidence that it hardly matters. This is truly a novel for the twenty-first century.

Slow but good
Manifold is a series of books with big, visionary concepts, and Space is no different. This time the twist on the Fermi paradox has the aliens existing and actually quite near the Earth. Reid Malenfant investigates with a mysterious Japanese scientist Nemoto. The first contact is made and the truth starts to unfurl...

As I said, the ideas are big - seriously big. The flow of the story isn't always fast enough, it all gets a bit too slow at times. Still, one has to admire Baxter's vision and while parts of the book were slightly boring, the whole of the story was definitely captivating enough to get me through the slower bits.

Manifold: Space offers an interesting what-if scenario of the future of humankind in a world that has extraterrestrial life.

Technically great, but ultimately cold and depressing
Manifold: Space is a very frustrating novel.As with its predecessor, Manifold: Time, it is brimming with great hard science surrounding a very good premise.In the Manifold series, Baxter gives detailed and extended perspectives to "Fermi's paradox".Fermi articulated that, due to the huge extent and age of the universe, either life on Earth is completely unique and we are alone in the universe (the basis of in Manifold: Time), or life must be everywhere and we simply have yet to discover it (as espoused in this novel).The paradox with the 2nd view is that if life is everywhere, the age of the universe implies that we cannot be the first cognitive, noise-making intelligence; why, then, haven't we found evidence of this other life?Baxter's answer to this paradox is quite interesting:he ties up the multiple story threads of Manifold: Space with a good ending.

Unfortunately, problems Baxter had with characterizations in Manifold: Time come to full flower in Manifold: Space.This novel is too seriously flawed too make it an overall enjoyable read.

There is a lack of any kind of human "community" in Manifold: Space.The various astronauts/space discoverers that are central to the story are all unhappy loners.They go into space alone, seemingly unaided, and appear to have no friends or colleagues.The first astronaut to set foot on Venus, as one example, never communicates with anyone beyond a single individual on earth, and that individual is also a loner.Baxter often beautifully elucidates the technical side of space exploration, but appears to have no clue as to the human elements. Manifold: Space is a cold unhappy story:interesting science populated by lonely, depressed individuals.

MANIFOLD ZZZZZZZZzzzzzzz
Let me start by saying I am an avid Sci-Fi reader, the longer the book the better. I believe a good story takes a long time to tell ala Harry Turtledove. So when I read the plot and saw the book size, I thought I was in for a good read. I could not have been more incorrect. Here are my bullet points:

1) Excellent plot idea, but I think Baxter forgot what is was after the first 12 pages. I have never been a part of such a plot trainwreck as this. Its like he compiled a dozen different stories into one without any of them going anywhere.
2) Character developement was horrendous. He needs to take a lesson from Harry Turtledove or Kevin Anderson. Those guys can present twenty or more characters, make you remember and love each one, this guy can't do it with two.
3) Science. About the only thing this guy can get across, but its not involving, its like reading a NASA technical paper, done with lots of coffee.
4) Writing. He needs to take a lesson from Jack McDevitt. You need to make the book an incredible journey independent of the ending. I begged for the ending just to finish the book. If Nemoto was one thousand years old, I felt like it half way through the book. Baxter's writing style is cold and easily forgotten.

The plot to this review, skip this book. It goes everywhere and nowhere all at once. I implore Baxter to work on his writing style and present more concise plot lines, eliminate all the extra about prehistoric men crapping on themselves on Jupiter's moon and stick to the plot. The editors need to quite screwing around and edit these books. Like previous readers, the book could have been cut in half... ... Read more

Product Description

Giving new meaning to the term "fast food"

Rest-stop grade F meat patty? Nah. Nuggets of reconstituted poultry bits? Pass. Deep-fried fish discus? No, really, thanks all the same.

It's time to bid farewell to the roadside meal as you know it. Nearly twenty years ago, Chris Maynard and Bill Scheller opened the world's eyes to the beautym of car-engine gastronomy in the original Manifold Destiny. And now that another generation of both drivers and eaters has emerged, the cult classic is due for an overhaul. In this shiny, spanking-new edition, learn how to make s'mores in your Scion, poach fish in your Pontiac, even bust out a gourmet snack from under the hood of your Escalade.

With step-by-step diagrams, crowd-pleasing recipes, and thorough instructions, now you can turn your car into a kitchen without ever crossing any golden arches. Hilarious, bizarre, and ultimately (seriously!) useful, Manifold Destiny is and always will be an unparalleled original. So, slap a ham steak under the hood of your car, hit the gas, and drive until you reach delicious -- which is in approximately fifty miles, depending on traffic. ... Read more

Customer Reviews (20)

Nice, quick service
Was a gift for a friend who had a copy years ago and didn't think it was even in print any longer.

Unique little book, humorous a kind of a "'60s" item.

Manifold Cooking
I got this as a gag gift for my son-in-law's birthday, along with another one on roadkill cuisine.My children and I experimented with cooking in the rear window (lined with foil), a box painted black and lined with foil, etc., all of which he found very funny.

It's not his birthday yet, but I hope that when he does get the book, he enjoys reading it as much as I have.Shoot - some of the recipes even sound pretty good!If I drove much, I might try a few of them.:oD))

Great Fun
This book is written with humor along with ACTUAL recipes ie;
IMPRESSA VEAL IMPREZZA = cooking time is listed as 30 to 50 miles.

The recipe for a meal cooked on a foreign car's engine has a comical cooking time. =
Nifty Naffta Nachos"Distance about 60 miles or forever, Depending on the presence of the Federales".

It's well worth the price, if for nothing else than a conversation piece. I purchased three extras as gifts and all recipients loved them. We read them together over the phone with each person having their favorite parts. Three of us are planning to try the recipes!

Terrific Redneck Gift
I love this book.I added a silicone mitt, some aluminum foil, a canteen and an all-in-one camping utinsil kit to complete the gift.It got lots of laughs and all involved swear that they plan to try out some of the recipes this summer.

Great book, a must-buy!
Read Manifold Destiny review years before it originally went out of print, and very happy to be able to get it again! A great gift for any car buff or gear head that you know. ... Read more

Product Description

Customer Reviews (31)

newspaper quality
I'm giving this review not so much to discuss the contents of the book, which I think is very good and would give*****, but the production quality of this edition which is truly appalling.As an undergraduate I used the first edition (OK, I admit I'm retired) which was on nice semi-glossy paper with crisp printing and figures.This edition from Westview Press looks like it was faxed onto newsprint.The earlier edition is virtually impossible to find so you are probably stuck with this one.If however you can get the earlier one, it is worth paying extra to buy a book and not something that looks like a newspaper flyer.

Great book.If you're prepared that is.
In the preface Spivak says that this book is accessible to anyone that has had good courses in calculus and linear algebra.While this is true, the term "good" needs a little clarification.If you've taken a two semester calculus sequence using the "cook book" approach, you will be grossly under-prepared for Spivak.Ideally you'd have taken one of the many theoretical calculus with honors courses offered at good universities or a nice undergraduate analysis course.I'd suggest preparing by means of Spivak's own book, Calculus, barring that you'll do well with Introduction to Analysis by Maxwell Rosenlicht, Understanding Analysis by Stephen Abbott, or, if you want to be old school, the first seven chapters of Principles of Mathematical Analysis by Walter Rudin.All the aforementioned books (except for Spivak's Calculus) cover topology and continuous functions at a more leisurely pace than Spivak.For the linear algebra I'd suggest one of Finite Dimensional Vector Spaces by Paul Halmos, Linear Algebra Done Right by Axler, or Linear Algebra by Peter Lax.The book by Lax is a wee bit more advanced and may require more maturity.Also the book by Axler downplays the role of matrices, focusing instead on linear maps as separate entities that can be expressed by means of a matrix given a basis.While this is a good approach, I'd suggest making an effort learn how to play with matrices.A lot of properties of differentiable functions from R^m to R^n can be handled with matrix tricksIf you work through one of the three calculus books and one of the three analysis books I've recommended you'll be in great shape for Spivak.It must be said that the exercises in Spivak are challenging and many are proofs of nontrivial theorems.It's hard work and requires significant creativity on the part of the reader.Of course, this makes the book ideal for training up new mathematicians.That's what the book is for and it is excellent at that; which is why I've given it five stars.

More Advanced Than It Claims
Be warned:Spivak is speaking a different language from any I was taught.I have spent hundreds of hours on this book, and have filled three binders with notes on the first three chapters.If you are using it in the right manner, it is a fantastic book;otherwise, it is a nightmarish exercise in frustration.Expect to rewrite the textbook in your efforts to understand it;for some, this is a plus.It is an excellent book for inspiring effort and creativity, but it is dreadful if you do not have outside support.It is NOT a primary text by any stretch of the imagination.

The first blow against the book is its claim that "only a term of linear algebra, a nodding acquaintance with the notation of set theory, and a respectable first-year calculus course" are required.This is enormously demoralizing to the student who has these prerequisites and yet finds Spivak incredibly obscure;but it IS obscure.The problem here is partly one of miscommunication:it turns out that Spivak has written his own "Calculus" textbook, based on his own notions of what such a course should contain.Having examined and read part of that text, I would describe Spivak's "Calculus" as a challenging introductory analysis textbook with a deep review of calculus.It is absolutely not a book to learn calculus from in the first place, unless one is prepared to spend five times as much time as it should take, and then to relearn the entire subject in order to be able to communicate with everyone else.

The second blow against it is the notation:Spivak has invented his own.Despite having had calculus, and being a successful calculus tutor, I found it nearly impossible to recognize the basic rules of differentiation in his encryption.The situation was bad enough that I felt compelled to devise my own hybrid notation to translate back and forth between the style used in Spivak and the style I have seen everywhere else in my studies, and to add clarity at a few crucial confusing points.

The third blow is that there is no clear indication of what tools are expected to be available to the student.I ended up taking breaks to devise and prove large sets of lemmas and theorems in topology and analysis just to make the problems more feasible.I suspect that that was the intent--but one might have said so.

The fourth problem is also part of its appeal:it is incredibly concise.This means that there are next to no examples, and huge amounts of detail have been skipped.Far too much has been skipped in places.The Implicit Function Theorem proof in particular was a nightmare;I had to completely rewrite it.

This book is most properly used as a challenge problem set for fairly advanced mathematics students, with guidance, to inspire huge amounts of work and creativity.It is least properly used as a primary text, an introduction, or a means to learn anything rapidly.

disappointing
What a bunch of hype on this book.It's format is so concise as to be practically useless. It seems to be a bunch of class notes cobbled together.Mulitvariable calculus in a book that is only 5 inches across?What a dis-service to the subject.It can barely contain the full definitions for tensor products. Just silly.Lacking examples or much of anything for the reader to bridge the gap his terse prose leaves. A Font so small that indices are lost on some equations. It is no Differential Geometry. The difficulty is not within the subject matter, but with it's meager presentation.Not at all the quality you would expect from this author.God, I hope this isn't "The" book on calculus on manifolds. Not to mention the typos.I never knew that f(a)= f(a)+ f'(a)*h...oh wait it isn't, but f(a+h) is!Iguess it was tough proof reading a hundred pages with 10 sentences on each page.Get the cheapest version you can.

Tip of the iceberg...
This book would serve well as a self-study introduction to smooth manifolds for a student just finishing a high school calculus course. For those serious about the subject, pick up John M. Lee's "Introduction to smooth manifolds." ... Read more

Product Description
This book is an introduction to manifolds at the beginning graduate level. It contains the essential topological ideas that are needed for the further study of manifolds, particularly in the context of differential geometry, algebraic topology, and related fields. Its guiding philosophy is to develop these ideas rigorously buteconomically, with minimal prerequisites and plenty of geometric intuition.

A course on manifolds differs from most other introductorymathematics graduate courses in that the subject matter is often completely unfamiliar. Unlike algebra and analysis, which all math majors see as undergraduates, manifolds enter the curriculum much later. It is even possible to get through an entire undergraduate mathematics education without ever hearing the word"manifold." Yet manifolds are part of the basic vocabulary of modern mathematics, and students need to know them as intimately as they know the integers, the real numbers, Euclidean spaces, groups, rings, and fields.

In his beautifully-conceived Introduction, the author motivates the technical developments to follow by explaining some of the roles manifolds play in diverse branches of mathematics and physics. Then he goes on to introduce the basics of general topology and continues with the fundamental group, covering spaces, and elementary homology theory. Manifolds are introduced early and used as the main examples throughout.

John M. Lee is currently Professor of Mathematics at the University of Washington in Seattle. In addition to pursuing research in differential geometry and partial differential equations, he has been teaching undergraduate and graduate courses on manifoldsat U.W. and Harvard University for more than fifteen years. ... Read more

Customer Reviews (9)

You'll learn manifolds
This book is a clear way to learn manifolds. In my opinion Lee is a great author with a clear knowledge of diferencial geometry. It worth to make this trip.

The printing is not up to the standard of the writing
By all accounts, this and Dr. Lee's other two books on manifolds are exceptionally well-written.But my copies arrived from Amazon this week, and, unfortunately, Amazon and Springer have decided to replace the crisp offset-printing of earlier printings by lower quality digitally-printed versions, probably as a cost-cutting measure.

If you care about how books look, I'd suggest trying Amazon marketplace or small retailers elsewhere to increase your odds of getting a superior copy from an earlier printing.

At the other end of the spectrum
The only other books I have seen that deserve the title of both a reference and a textbook are by Lang. That being said, the exposition is at the other end of the spectrum with the goal being to teach the reader and not develop topology from the foundations in excruciating detail. The strength of the book (the same applies to the other titles by the same author) is that there is no compromise in rigor and the user-friendliness of the book relies upon the motivating discussions spread throughout the text. The author seems to have a long teaching experience and so he wisely advises the reader to work through the exercises. There is a pitfall with Lee's books: because the explanations are so lucid and intuitively satisfactory you might fool yourself that you know the material but this is not the case until you solve most of the problems.

Excellent book
I used Lee's 'Introduction to Smooth Manifolds' & 'Introduction to Curvature' for a few months, and I felt like it would be a good idea to complete the collection and acquire some more knowledge about topological manifolds using this book.

Overall, I think that Lee's 'Introduction to Topological Book' is an excellent book, as it is one of the few books that give both a profound intuition to the geometry of the subject, supplemented by rigorous proofs.
In my opinion, the book has only two disadvantages:
1) Not enough concrete examples.
2) It should has covered a bit more material, such as CW complexes.

All together, HIGHLY RECOMMENDED! The first chapter, giving some motivation to study manifolds is awesome!

A welcome text to my collection
I picked this book mainly because a friend recommended this whole series to me.While I cannot say this book would make a great introduction to point set topology (I think Munkres is still the best for that), it has all that one would want to get going with manifold theory.What I liked most about this text is probably the rigor.This text will motivate the topics and give rigorous proof to many theorems.There are also many good examples to illustrate his point.I'd recommend this book, and the follow-up text "Introduction to Smooth Manifolds" to anyone interested in graduate level mathematics.Since the two texts will likely cost you less than $100, they'll make a nice addition to your math library. ... Read more

Product Description
Intended for upper undergraduate or beginning graduate students, this book introduces students to the modern theory of manifolds. Assuming a basic knowledge of the differential geometry of curves and surfaces the focus is on differentiable manifolds and the study of Riemannian manifolds. The book concludes with applications of manifolds to physics. Exercises at the end of each section and appendices on topology and linear algebra make this book ideal for self-study or as a textbook. ... Read more

Customer Reviews (1)

Nice! Very clear, concise, rigorous reader-friendly introduction to differential manifolds
Lovett provides a very nice introduction to the differential geometry of manifolds useful for self-study.It is very clearly written, rigorous, concise yet reader-friendly.The difficulty level is midway between O'Neill's Elementary Differential Geometry, Revised 2nd Edition, Second Edition and Tu's An Introduction to Manifolds (Universitext) (Volume 0).

The pace is nice.As you can see in more detail from the "search inside this book" function: Ch. 1 Analysis of Multivariable Functions [pp. 1-36] provides some background math; Ch. 2 [pp. 37-78] Coordinates, Frames, and Tensor Notation discusses some more applied topics needed for physics applications; Ch. 3 Differential Manifolds [pp. 79-124] and Ch. 4 Analysis on Manifolds [pp. 125-184] discuss essential standard topics including differential maps; immersions, submersions and submanifolds; vector bundles; differential forms; integration and Stokes' Theorem;Ch. 5 [pp. 185-248] provides an introduction to Riemannian Geometry, including vector fields, geodesics and the curvature tensor; and finally Ch. 6 [pp. 249-294] provides very brief discussions of some applications to physics including Hamiltonian mechanics, electromagnetism, string theory and general relativity.

My main gripe is that there are no answers to problems, which detracts from its value for self-study (but to fill that gap, cf. Analysis and Algebra on Differentiable Manifolds: A Workbook for Students and Teachers). This is especially annoying because Lovett refers to answers to some problems in his mathematical exposition, e.g., on p. 234 (section 5.4.1), he refers to problem 5.2.17 on page 217 in his discussion of connections that are not symmetric; moreover answers to some exercises depend on material in other problems, e.g., the answer to problem 5.2.17 refers to problem 5.2.14.This is a common practice I dislike because it seriously degrades from a book's value for self-study. It could well be that one star should be deducted for this despicable practice.Nevertheless, I have given it 5 stars because I like the fact that it covers Riemannian Geometry (including an exposition of Pseudo-Riemannian metrics in section 5.1.4 and 5.3.3) and in section 6.4, a short introduction to general relativity but mostly because it's the only book I know that can help one make the leap from very elementary books like O'Neill's Elementary Differential Geometry, Revised 2nd Edition, Second Edition, Pressley's Elementary Differential Geometry (Springer Undergraduate Mathematics Series) or Banchof and Lovett's Differential Geometry of Curves and Surfaces to graduate level books like Tu's An Introduction to Manifolds (Universitext) (Volume 0), John Lee's Introduction to Smooth Manifoldsor Jeffrey Lee's massive [[ASIN:0821848151 Manifolds and Differential Geometry (Graduate Studies in Mathematics), all of which I also recommend after Lovett.

All in all, this text is a welcome addition to the many books on differential geometry because of its refreshing, "no nonsense" clarity, rigor and conciseness as well as the various topics covered. ... Read more

Product Description
“ONE OF THE BEST SF WRITERS IN THE BUSINESS . . . [Manifold: Origin is] filled with marvelous scientific speculations, strange events, novel concepts, and an awe-inspiring sense of the wonders of the universe.”
–Science Fiction Chronicle

In the year 2015, astronaut Reid Malenfant is flying over the African continent, intent on examining a mysterious glowing construct in Earth’s orbit. But when the very fabric of the sky tears open, spilling living creatures to the ground and pulling others inside (including his wife, Emma), Malenfant’s quest to uncover the unknown becomes personal. While desperately searching to discover what happened to the woman he loves, Malenfant embarks upon an adventure to the very fount of human development . . . on earth and beyond.Amazon.com Review
Award-winning author Stephen Baxter turns to the origin of species in this final novel of the Manifold trilogy. Reid Malenfant and Emma Stoney are flying over Africa when a new moon appears in the sky--and Emma disappears. She finds herself on the Red Moon with people resembling human evolutionary ancestors, with whom she must learn to live in order to survive. On Earth, Malenfant teams with Japanese scientist Nemoto on a desperate rescue mission that leads to greater questions about the origin of the alien moon, and ultimately of humankind.

Because the Manifold novels take place in alternate universes, Origin works well as a stand-alone read.Baxter effectively explores how modern humans and their ancestors might be thoroughly alien to one another, but the book is more focused on thoughtful scientific speculation than in-depth characterization. However, readers who are swept away by novels of cosmic scope and compelling imagination will find Big Idea science fiction at its best. --Roz Genessee ... Read more

Customer Reviews (30)

Caveman health problems
I enjoyed the first two books of the Manifold trilogy, but this one was horrid. There is more talk about various primate digestive problems than deep sci-fi or philosophical issues. Absolutely pointless.

Full of grand themes, but a disappointing end to the series
"Origin" is the third and final book in Stephen Baxter's Manifold sequence, exploring the conundrum known as the Fermi Paradox: if alien intelligences do exist, why don't we see evidence of them all around us? As in the first two books in the series - "Time" and "Space" - it follows the experiences of Reid Malenfant, a washout NASA astronaut, and his wife, Emma Stoney.

When a mysterious blue portal appears in the sky over Africa, it sweeps people from the Earth - including Emma - before abruptly vanishing. At the same time the moon is replaced in the sky by the Red Moon, and Malenfant, convinced that this is where his wife has been taken, lobbies NASA for a one-shot mission to bring her back, and to explore this strange new world. What he and Emma are surprised to discover is that the Red Moon is home to all manner of hominids - proto-humans from Earth's past, some of which were thought to have been extinct for millions of years. But who or what gathered them in this one place, and how?

While the book begins dramatically, it quickly becomes clear that this instalment lacks the vision and imagination of either "Time" or "Space", or even of the related short story collection, "Phase Space". Much attention is devoted to the viewpoints of the hominids themselves, but since these tend to revolve around the basic activities of survival, there is little to hold the reader's attention, and these sections soon prove tiresome. As a result the narrative progresses slowly, and with little sense of direction. Far more interesting is the story of the technologically superior super-beings known as the Daemons, by means of which Baxter explores how humans might have evolved under different circumstances.

Compared to the first two books, which by their nature allowed Baxter to roam free through space and explore many different environments, "Origin" feels rather claustrophobic, limited to the single setting of the Red Moon and a relatively small cast of characters. As for the final conclusion - the long-awaited resolution to the Fermi Paradox - this feels much like a rehash of themes explored in the earlier two volumes. It is clear that there are big ideas at the centre of the book trying to express themselves: ideas regarding the origin of our own species, the concept of human nature and the question of what makes us different from the animals, and (above all) the role of contingency in the universe. Perhaps a bigger stage was all that was needed.

Entertaining but ultimately disappointing, "Origin" is perhaps the weakest of the Manifold series. To his credit, however, Baxter has never been one to shy away from the big questions, and just as in "Time" and "Space", he continues in this novel to push the boundaries of science fiction and what it can do. Thus even if the final product doesn't quite come off, there is nevertheless much to be admired in his attempt.

Disgusting
The story is imaginative and well written. There are too many unnecessarily disgusting and violent scenes. After the third graphic and explicit baby-murder I just couldn't read any more.

Typical Stephen Baxter
Another good book in the loose trilogy of the manifold series.This one investigates associations between different levels of humanity on the rungs of evolution. As if they could somehow interact between the ages with some interesting manifestations developed to make points.
Right up there, almost, with Evolution, which is my favorite for his science, what if style, while of course Titan showcases his ability to create characters to bleed over. I haven't read all of his books, something that I'm remedying presently, but one of my top 10 authors since the release of Titan.
heartedly recommended with no real reservations.

Of the origins
Manifold is a series of three books. They're not a sequence, actually, as they describe parallel universes. The main character are the same, but the world they live in is different. Origin presents us a world where the good old Moon is replaced by a large red moon. As it happens, the main character, Reid Malenfant, loses his wife Emma on the new moon and has to rescue her.

Emma finds the new moon inhabited by various hominid species. Baxter offers us an interesting view to the life of different hominids, with a point of view of the hominids themselves and humans living with them. It's interesting, but it can also get slightly tedious - this is one long-winded book, with a plot that's a framework for all sorts of neat stuff Baxter wants to present.

But it works, for me at least, because even though I began reading book with some doubts, I soon got sucked into the events. It gets quite interesting and Baxter has some pretty wild ideas there. This book isn't for everybody (that is easy to see from the Amazon reviews, many of which give just one or two stars), but if you enjoyed the other Manifold books, this one is worth reading. ... Read more

Product Description
Proceeds from general to special, including chapters on vector analysis on manifolds and integration theory.
... Read more

Customer Reviews (15)

Can't get over how good this book is.
As a physics-math major, I have never come across such a perfect book to start differential geometry. I buy a lot of Dover publishing books because of their cheapness, but this one is probably my most valued geometry book. No other book has been this terse and this clear at the same time.

This book provides a solid foundation for everything it does without assuming your understanding of anything before-hand.

For example, physicist-geared introductions tend to hide a lot of the real topology and linear algebra behind the subject. Almost every other book I've read assumes knowledge of dual spaces for vector spaces, and just gives a quick definition. This book does not do any of that. It gives clear mathematical details and motivation to go with them.

Don't let the book's low price make it appear to be low quality.

Great Primer on Differential Geometry
I bought this as a primer for reading General Relativity and did not give five stars for a few reasons:
no symbol index, the intro on Topology is almost worthless and I would add a few more required subjects before bothering to tackle this besides the ones suggested by the authors including introductory Linear Algebra and Topology.My other complaint is that there are only 21 small diagrams which is kind of disappointing considering the highly geometrical nature of the subject.

A solid text on manifolds
I have some reservations:
1) no notation/ symbols page
2) uses the Klein Map but doesn't give the name
3) no clear affine and projective manifold classification
But in contrast it has good handling of diffeomorphisms and integrations
on manifolds.
I bought it to the Weeks space manifold, M003[3,-1], and the polyhedron forms that Weeks uses. It isn't really
of use in that more modern area of manifold theory either.
The Thurston space manifold, M003[2,-3], isn't covered.
The book is a good introductory text that I wish I would have had 40 years ago!

Tough for self-study
I have been using this book to study differential geometry for many years - a little bit at a time. This book is a fairly complete introduction to the subject. However, it does a poor job motivating and explaining the subject. I found it necessary to supplement with several other texts to really get a good grasp on the material in the book. A number of times, I have picked up something in another book and have gone back to this book and realized that I had not "gotten it" the first time through. If the book had more examples and concrete calculations it would go a long way to clarifying the material.

I would recommend getting a book like Guggenheimer's Differential Geometry and reading it first. This book then does a good job of generalizing the ideas to many dimensions.

Fantastic
Great book. Very clear explanation of what manifolds are and how to use them. Also, the prerequisites are minimum. You'll get a lot of mileage out of this book if you have a semester or so of advanced calculus and some topology and linear algebra under your belt. I wish there was there was a chapter on bundles because I would love to read what these guys have to say about them. Very readable. Definitely a good buy. ... Read more

Product Description
If Christians are part of the one body of Christ, how do we account theologically for the multiple expressions of our common faith? If God is ultimate truth, why is it so difficult to agree on issues related to truth? Must we sacrifice a commitment to truth in favor of a pragmatic unity in the church? Or must we hold on to our perception of the truth at the expense of fracturing the church? For John Franke, truth versus unity is a false dichotomy. In this provocative yet thoughtful book, he argues that orthodox and biblical Christian faith is inherently pluralist, and that this diversity, far from being a problem that needs to be overcome, is in fact a blessing from God and part of the divine design and intention for the church. Suggesting that Christians should affirm the reality of ultimate truth, but cautioning humility regarding our grasp of it, Franke sets forth a relational theology in which the many expressions of revealed truth Christ, the Holy Spirit, and Scripture, along with a diverse church together witness to the expansiveness of the one God. ... Read more

Customer Reviews (2)

Engaging, But Unfulfilling
John Franke has written an engaging book, and the subject matter is eternally relevant and always controversial: truth.In Manifold Witness, Franke is arguing for an ever-present plurality within Christian truth claims.Throughout his work, Franke engages questions concerning the nature of truth, the nature of God, explains deconstruction as a vital practice for Christian theological reflection, and adds a proposed approach to theology based on his concept of "manifold witness."From start to finish, this is an interesting read that takes up many important contemporary concerns.

Franke raises some helpful questions, such as whether a historic Christian faith exists.Relying on the work of Andrew Wall, Franke posits what the Christian faith would look like overtime to a space alien researching human religious culture.Focusing on Christianity in particular, Franke asks if the alien would recognize any coherence at all, and if the Vincentian rule of the faith, that which has been believed "everywhere and by all," could be successfully applied.Franke believes it could not.While there would be some similarities found within the Christian faith as it has been professed across time, Franke believes there is more diversity than there is unity, and as such can be considered an evidence for the plurality of truth.While Franke's scenario is plausible, I humbly disagree.Plurality of expression, and particular emphases on certain facets of Christian doctrine and practice over time to meet specific needs and specific cultural situations does not automatically negate a unified "deposit" of Christian faith that has been constant across traditions and across time.Plurality of expression does not exclude unity and singularity of truth.

Franke's work also is Trinitarian, and for this reason is to be commended.The resurgence of Trinitarian theology has been good news for Christianity, and has led to deepening reflection in particular on the social nature of the Godhead, which is Franke's focus here.Franke extends the sociality of the Trinity to the loving, missional impulse of the Father, Son, and Holy Spirit who are now at work to bring to full expression the redemption that has been made possible through Jesus Christ.This presentation of the Trinity does have some strengths, as it can serve as grounds for the ministry of the local church and the imperative for engagement and service to the world.However, throughout Franke's discussion of the Trinity, I couldn't shake the sense that the plurality of God was being stressed over and above the unity of the Godhead in a way that is unwarranted.It is as though the commitment to truth as plural was driving the assertion that the Trinity's plurality takes precedence over the Divine Oneness of the Father, Son, and Holy Spirit.

In the final analysis, I can't help but wonder how Franke's presentation in Manifold Witness is able to discern which expressions of the plurality of truth can be upheld and considered authoritative for the body of Christ.While his critiques of cultural superiority (the belief that one particular cultural expression of Christianity can be universalized over all others) are extremely helpful and humbling, for the Church as a whole, I cannot see how such an approach to truth allows for discernment between what, in fact, is true and false.It seems that in the end, the plurality would be favored over the unity, for the ground for unity is evasive.

This is an interesting read in contemporary theology, and those with a postmodern bent might particularly find it compelling.But in the end, I found the book a good read that left me a bit empty, hoping for more.While I recognize that mystery is a facet of Christian faith, I'm seeking for some firm ground to stand upon while engaging in contemplation of Christian truth.The search continues.

The truth about truth
While the topic of truth gets a lot of air-time these days, few actually take the time to define what they are talking about or move beyond critiquing the "other side."Franke though stays true to an evangelical affirmation of truth while at the same time thoughtfully engaging with the reality of pluralism.His nuanced approach to the discussion doesn't rubber-stamp any extreme, but admits the complexity associated with faith and truth.And for that, I found his work to be refreshing.He admits upfront that "the expression of biblical and orthodox Christian faith is inherently and irreducibly pluralist" (7).But this isn't an in-your-face assertion that must be swallowed whole; it is instead the idea that the whole book seeks to unpack and explore.With a faithful commitment to scripture and a tender compassion for the reader, Franke demonstrates how pluralism is not something to be feared or fought but is instead simply a beautiful intrinsic aspect of not just our faith but all creation.

I appreciated how Franke in his discussion of truth quickly moved beyond the absolute and relative dichotomies.Neither accurately represents truth as the first tries to commoditize it for the sake of power and the second deny it in the name of tolerance.Pluralism and truth are far more complex than the extreme camps allow us to admit.Our world is diverse, as is our faith.And Franke rightly points out that culture and our faith is always changing, God never leaves us where we are at, but is constantly transforming us with the gospel.The constant renewing of our minds allows us to faithfully claim traditions in the church as well as celebrate the new things God is doing.The celebration of plurality affirms the "importance of multiple perspectives in the apprehension and communication of truth" (40).Just as The Father, Son, and Spirit are one even as they are different, the church can be one while living fully into our own diversity.

I also was grateful for Franke's assertion that we can never let our particular cultural setting trump our commitment to truth.We are situated in culture, but when we start to assume that our cultural habits are the only way to present truth, we are in fact limiting God and truth.Scripture and God cannot be subject to cultural assumptions, but must be celebrated in their plurality. Similarly, we should remember that God doesn't seek to assimilate the Other and make us all the same either.Franke brilliantly reminds us that we can be silencing God when we do not listen to voices that might not fit our accepted cultural theological norms.He writes, "theology is not a universal language.It is situated language that reflects the goals, aspirations, and beliefs of a particular people, a particular community" (94).If we are to affirm the plurality that God affirms, we must thoughtfully seek out the diversity of theological voices. This was a poignant wake-up call for me as I too often only listen to the voices of those similar to me.I need to be striving to affirm God by affirming the truth of the many legitimate enculturations of the faith.

Manifold Witness is accessible, but it is also challenging.Franke goes places that others have avoided - not for the sake of controversy, but out of a deep desire to be faithful.His commitment to loving and serving God is apparent on every page of this book making his exploration of the plural nature of truth a gift to the Christian community.I highly recommend this book not just for those caught up in the discussion of truth, but to all Christians eager to celebrate our expansive God in the full diversity of his church.
... Read more

Product Description
This text is designed for a one-quarter or one-semester graduate couse in Riemannian geometry. It focuses on developing an intimate acquaintance with the geometric meaning of curvature and thereby introduces and demonstrates all the main technical tools needed for a more advanced course on Riemannian manifolds. The book begins with a careful treatment of the machinery of metrics, connections, and geodesics, and then introduces the Riemann curvature tensor, before moving on the submanifold theory, in order to give the curvature tensor a concrete quantitative interpretation. The remainder of the text is devoted to proving the four most fundamental theorems relating curvature and topology: the Gauss-Bonnet Theorem, the Cartan-Hadamard Theorem, Bonnet's Theorem, and a special case of the Cartan-Ambrose- Hicks Theorem. This unique volume will especially appeal to students by presenting a selective introduction to the main ides of the subject in an easily accessible way. The material is ideal for a single course, but broad enough to provide students with a firm foundation from which to pursue research or develop applications in Riemannian geometry and other fields that use its tools. Of special interest are the "exercises" and "problems" dispersed throughout the text. The exercises are carefully chosen and timed so as to give the reader opportunities to review material that has just been introduced, to practice working with the definitions, and to develop skills that are used later in the book. The problems that conclude the chapters are generally more difficult. They not only introduce new material not covered in the body of the text, but they also provide the students with indispensable practice. ... Read more

Customer Reviews (8)

geometric understanding of QM
I'm not qualified to give the review, but this is one of the top books in the quantum systems engineering group that John Sidles recommends and reads.Plus, anything with Riemannian in the title recalls Cryptonomicon which can't be all bad.

Do Carmo's is better
I've taught an introductory differential geometry course from Lee's book, and in retrospect Do Carmo's "Riemannian Geometry" would have been a better choice. To be fair Lee does masterful job introducing basic concepts from curvature to Jacobi fields, but here are a few things I disliked. The book assumes working knowledge of smooth manifolds and Lie brackets, while many students need review of the former, and know nothing of the latter. Lee doesn't give enough examples beyond constant curvature spaces: there is virtually no mention of warped products, Riemannian submersions, Lie groups, or homogeneous spaces. Exercises are few, unmotivated, and their difficulty is in stark contrast with the easiness of the main text. I feel Do Carmo's book is superior in all respects, and last time I checked it was not much more expensive.

The printing is not up to the standard of the writing
By all accounts, this and Dr. Lee's other two books on manifolds are exceptionally well-written.But my copies arrived from Amazon this week, and, unfortunately, Amazon and Springer have decided to replace the crisp offset-printing of earlier printings by lower quality digitally-printed versions, probably as a cost-cutting measure.

If you care about how books look, I'd suggest trying Amazon marketplace or small retailers elsewhere to increase your odds of getting a superior copy from an earlier printing.

Excelent Book!
This book is great.An excellent book for anyone with a little background in mathematical formality, i.e. analysis, topology or differential geometry.

As always
Prof. Lee sets the norm of mathematical exposition. I would give it 5 stars if it were more comprehensive. There is so much to say about Riemannian manifolds and it would be a pleasure to see them under the light the author sheds on such subtle concepts. One very nice feature of the book that underlies its structure is that it uses four theorems - pillars of Riemannian geometry as a guide of what should be included. This approach, besides improving considerably the organization of the book as compared to other books on the subject, it also motivates the reader who now has a target in his mind, namely the proofs of these important theorems. It is really nontrivial to introduce people to mathematical areas as broad as Riemannian geometry. Also, useful suggestions are given in the preface for further reading. ... Read more

Product Description

Manifolds, the higher-dimensional analogs of smooth curves and surfaces, are fundamental objects in modern mathematics. Combining aspects of algebra, topology, and analysis, manifolds have also been applied to classical mechanics, general relativity, and quantum field theory.

In this streamlined introduction to the subject, the theory of manifolds is presented with the aim of helping the reader achieve a rapid mastery of the essential topics. By the end of the book the reader should be able to compute, at least for simple spaces, one of the most basic topological invariants of a manifold, its de Rham cohomology. Along the way the reader acquires the knowledge and skills necessary for further study of geometry and topology. The requisite point-set topology is included in an appendix of twenty pages; other appendices review facts from real analysis and linear algebra. Hints and solutions are provided to many of the exercises and problems.

This work may be used as the text for a one-semester graduate or advanced undergraduate course, as well as by students engaged in self-study. Requiring only minimal undergraduate prerequisites, Introduction to Manifolds is also an excellent foundation for Springer GTM 82, Differential Forms in Algebraic Topology.

Customer Reviews (5)

A graduate physics student
If you are in a technical field of study and keep seeing words like, manifold, tangent bundle, vector bundle, lie groups ..., and want to know what they are, this book is a must. For anyone studying General Relativity, Quantum Mechanics or Quantum field theory, this book can clear up some of the mathematical confusion of these subjects.

This book covers the mathematics included in most upper level science courses that is not covered in the required math courses. This book is easy enough to understand to give any one in a technical field an edge. This book help me so much, that I was able to go and read more abstract math books.

This book should be required reading for anyone entering graduate level physics. If you are like me and really want to understand physics, or any of the sciences at a higher mathematical level then get this book.

Un-illuminating exercises, full of errata
I find this book to be a very frustrating textbook for my graduate-level differential geometry course.The exercises are un-illuminating verifications of simple problems, and don't really help me understand the concepts.The book is chock full of typos, many (but by no means all) of which are listed in an errata sheet available at [...].Among the most egregious is that in the section on point-set topology, the author defines a closed map as one which maps closed sets to open sets.

The book is not entirely without merit.The actual body of the text provides a reasonable introduction to manifold theory.Despite an extensive background in point-set topology, algebraic topology, and algebra, I still find the subject exceptionally confusing.

Clear and Solid Exposition
This is my favorite book on Differentiable Manifolds. After reading this book the reader will obtain a solid background on the following essential notions: Charts and atlas of a manifold; tangent vectors (as derivations); differential of a smooth function between manifolds; submanifolds and embeddings; quotient spaces; partitions of unity; vector fields; vector bundles; differential forms and de Rham cohomology. And on the road, the reader gets a gentle exposure to Lie groups, Lie algebras; and some basic notion of Category and Functors.

I found the following aspects of the book especially attractive:
> Clear style of writing: The author is the coauthor of the acclaimed "Differential Forms in Algebraic Topology". See the comments for that book. The clarity has not decreased at all.
> Bite-sized sections: The materials contained in each section is approximately equal to that of a 50-minute lecture. This helps readers who plan self-study.
> Right amount of topics: This is not an encyclopedia on manifolds. However, it does contain the ``absolute must'' one should know about manifolds. And it does such a good job in presenting it, the reader will be left with a solid understanding on those essential topics.

I first read this book as a Physics student and had no trouble reading it. I later switched discipline to Mathematics, and I know that this book has helped me appreciate the beauty of Mathematics. I thank the author for writing such an wonderful book.

An introduction it Manifolds
This is an excellent book. I wish that more books on advanced mathematics were written in this style. In contrast to most books on manifolds that tend to be very difficult for beginners to follow, Prof. Tu has made every effort to make this subject understandable to the nonexpert.

Greg Chirikjian
Professor, Mechanical Engineering
Johns Hopkins University

fills a gaping hole

i think there is a jump from ugrad analysis/alg/top etc to early grad school concepts. i didnt know category theory, i only had the flimsiest notion of a manifold, etc etc. and this book fills in that jump wonderfully. it does the right mix of analysis-differential topology-topology so that you can go read a book like bott and tu later (that's what it was designed for).
so im having a good time with it. ... Read more

Product Description
The year is 2010. More than a century of ecological damage, industrial and technological expansion, and unchecked population growth has left the Earth on the brink of devastation. As the world's governments turn inward, one man dares to envision a bolder, brighter future. That man, Reid Malenfant, has a very different solution to the problems plaguing the planet: the exploration and colonization of space. Now Malenfant gambles the very existence of time on a single desperate throw of the dice. Battling national sabotage and international outcry, as apocalyptic riots sweep the globe, he builds a spacecraft and launches it into deep space. The odds are a trillion to one against him. Or are they?Amazon.com Review
Leave it to the consistently clever Stephen Baxter to pull the old bait and switch. A story that begins as a hoary asteroid-mining tale, set in 2010 against the by-now familiar spiel of fulfilling humanity's pan-galactic Manifest Destiny, instead takes a bold, delightful ascent into a trajectory far more ambitious. To ensure its survival, humankind need not merely master the galaxy but also the flow of time itself.

Manifold: Time's would-be asteroid-miner-in-chief is bootstrap space entrepreneur Reid Malenfant, a media-savvy firebrand who's showed those crotchety NASA folks what's what with his ready-to-fly Big Dumb Booster, piloted by a genetically enhanced super-squid. But Malenfant's near-term plans to exploit the asteroids get diverted when he crosses paths with creepy mathematician and eschatologist Cornelius Taine. Applying Bayes's theorem and a series of other statistical do-si-dos, Taine convinces Malenfant that an inescapable extinction event--the "Carter catastrophe"--is nigh, and that even working to colonize the galaxy might not be enough to save humanity. The answer: build a Feynman "radio" to listen to the future and, by detecting coded quantum waves traveling back through time, divine the fate of human "downstreamers" and find the key to their survival. Space flight, time travel, and even squid negotiations ensue, while Earth is gripped in Last Days madness.

Once again, the award-spangled Baxter gives us sci-fi at its beard-stroking best, with an imaginative, audacious plot line that's firmly grounded in good science, reminiscent of Baxter's own excellent Vacuum Diagrams. --Paul Hughes ... Read more

Customer Reviews (113)

Profound Ideas
Baxter's 'Manifold Time' focuses on some truly profound ideas and mind-boggling time scales. Very deep.

Yes, the dialogue, plot, and characters are at times a bit thin. Overall the plot and writerly craft pick up during the second half. But the point of 'Manifold Time' is the science and the ideas, and sublime ideas they are. If you are just looking for a dumb, cheap thriller, this is not for you. But anyone who appreciates Carl Sagan or Michio Kaku and the accompanying deep thoughts of astrophysics and the universe should appreciate this book.

wasted potential
The beginning of the book was quite engrossing, but the plot kept wandering all over the place.When the story suddenly, and unexpectedly, started talking about some super-intelligent squid, I quit.Any interest I had in continuing the book was gone.

A Physics Fairy Tale - Undisciplined Writing - Cheesy
The characters were weak, but so was the story line. Too many times I had to suspend my understanding of physics to let the author plod to the next impossible description. For example, the author seems to think that reflected light travels at infinite velocity (or very much faster than c) whereas the light coming directly at you from the event is traveling at c. (Characters are watching an expanding sphere of light from an explosion, waiting for it to hit them.) The book has many such flaws. And the characters are equally flawed and thinly portrayed. I can't recommend this book to anyone expecting hard SF, unless you want to groan a lot.

I Got Bogged Down...
I am a fan of Stephen Baxter's. Vacuum Diagrams and The Time Ships were two of my favorite sci-fi books in the last ten years (at least among the Sci Fi I have read.) And I was looking forward to diving into a meaty trilogy of his that we could me reading for awhile. However whereas those two novel's took some fascinating contemporary science and built interesting conflicts and narratives on top of them, this book drowns beneath them.

Too often the action gets bogged down in a scene where one scientist or mathematician is standing in a room with one of the protagonists (who were neither) explaining some scientific principle or another which Baxter feels in imperative to the story. And just as the protagonists through one cliche or another express their confusion ("In English" - "X...tried to act like they understood." - "Malenfant tried to contain his frustrating confusion.") over and over and over again, so too was I squinting at the page and struggling to distill the important principles. Invariably the scientist or mathematician would sigh in patronizing frustration at the protagonist/me and simplify things...which they could have just done to begin with.

This happens over and over again to the point where I just got bored and ended up getting bogged down in this one for quite awhile. It's a pity because this past weekend I finally made a concerted effort to finish it and, where the first 250 pages were like a pushup drill, the last 150 were a lot of fun and I flew through them. In typical Baxter style, the story was elevated from interesting straightforward premises to questions about the very nature of the universe and what could be our place in it's present, beginning, and ultimate end. Even in the midst of the climax there was STILL that convention of the smart characters stopping to explain what was happening to the dullards in the story, but at that point the action had reached a level that I didn't care.

Even though I found this one excruciating at points I'm surprisingly still interested in the sequels, if only because I have no idea how this one could carry on. If you can soldier through the first half this one gets a hesitant recommendation.

confuses scale with depth
much less thancompelling characters. the truths described in the book make little sense and have other easier explanations. in short. big ideas but trite. ... Read more

Product Description
Diesel Dining: The Art of Manifold Cooking is for hungry truck drivers. Hard working people not having the luxury of being at home, and wanting something at the end of a long day resembling a home-cooked dinner. Diesel Dining: The Art of Manifold Cooking teaches you how to prepare good, healthy, hot, affordable meals. Diesel Dining: The Art of Manifold Cooking is guaranteed to save you thousands of dollars a year by removing the temptation and impulse buying of overpriced fast food products when you are famished and too exhausted to cook. Diesel Dining: The Art of Manifold Cooking offers a variety of recipes you can try cooking on your manifold. It also includes tips, tricks, stories, and trucking folklore.If you're a long haul trucker, and you want to enjoy a home-cooked, hot, meat and potatoes dinner at the end of your day's drive, then Diesel Dining: The Art of Manifold Cooking is the one and only cookbook you'll ever need. You'll never see Diesel Dining: The Art of Manifold Cooking in a typical homemaker's kitchen, since its premise is based on using your truck's diesel engine to cook your meals. Soon, this book will be dog-eared and stained with a variety of sauces, as it sits within easy reach by your interstate maps and daily log book. ... Read more

Customer Reviews (2)

so creative!
This book is great! The authors have figured out how to put food on the manifold of trucks to cook their dinner as they drive down the highway. Even if you aren't a long haul truck driver, and won't be able to cook as you drive, the recipes wouldn't be hard to adapt to a regular kitchen.

LOVE it!!
What a wonderful and unique book!My husband and I have tried some of the recipes and we are hooked! Try Smillin' Cec's Chunky BBQ Sauce! Much more than a book of recipes, it's very entertaining, and would make a terrific gift too! ... Read more

Product Description

This is the first book on multivariate analysis to look at large data sets which describes the state of the art in analyzing such data. Material such as database management systems is included that has never appeared in statistics books before.

Customer Reviews (6)

a truly modern treatment of multivariate analysis
Traditional graduate level texts such as Ted Anderson's focus on the multivariate normal distribution and its statistical properties.So out of that we get MANOVA, Hotelling's T square, linear and quadratic discriminant analysis, principal component analysis, Wishart distributions and canonical correlations.As other reviewers have said this book is quite different.You don't see those topics as chapters in this book.In fact most of these topics are avoided. Izenman finds that with large dimensional data sets that come up in practice these classical techniques do not work very well.So he takes a more modern and "nonparametric' approach.Color adds to the attractiveness of the book although often not essential to the graphical description of the data.

The book begins with exploratory data analysis and extends it to the realm of data mining.The ability to do analysis like this on large data sets comes from the amazing advances in computer speed. Several important concepts are introduced in intuitive ways including pattern recognition and machine learning, prediction error, cross-validation and bootstrap, and overfitting of models.

Chapter is again aimed at the practical by emphaiszing data structure and data bases and by introducing data quality issues including data inconsistencies, outlying observations (which becomes more complicated in multivariate analysis as many directions in a multivariate space can be considered extreme), missing data, and common to today's research data containing many variables but only a few observations such as gene expression on microarrays and satellite images.

But great ideas are not always modern. Izenman points to the curse of dimensionality, a concept coined by Richard Belman back in 1961.Chapter 3 on random vectors and matrices is the one place where the multivariate normal theory is explicitly covered.

Chapter 4 is truly nonparametric and covers multivariate density estimation with instructional examples sprinkled throughout the chapter.Chapter 5 deals with multiple regression a very important and common technique that is described in many texts.Izenman starts with some historical perspective going back to work on least squares by Gauss, Laplace and Legendre where the determination of planetary orbits were modeled circa 1800 and the work of Galton on heredity and regression to the mean in the 1880s and 90s.He gets to all the classical work but also discussion prediction error, the bias of the apparent error rate for a model estimate and the use of cross-validation and the bootstrap as ways to remove large biases in estimates.Again teh techniques are demonstrated with real examples.He discusses some reasonable biased regression approaches including ridge regression, principal components regression and partial least squares regression.Some of these techniques are new even to me (an aging statistician).Since practical problems often involve many potential variables of which some may be unimportant or highly correlated with others, practical regression analysis often use variable selection techniques.Izenman explains the methods and the associated controversies with them.He then introduces some modern approaches not seen outside the research literature including regularized regression (Friedman's general penalized least square approach and the Tibshirani's lasso and Brieman's garotte).He also devotes a whole section to least angle regression developed by Efron, Hastie, Johnstone and Tibshirani.

Chapter 6 generalizes to multivariate regression which includes MANOVA and MANCOVA.Chapter 7 deals with linear dimensionality reduction which includes the classical principal component analysis, canonical variables and canonical correlation and generalizations and then moves to the not so commonly treated topic of project pursuit.In several of the chapters including chapter 7 software packages are listed that can be used to implement the techniques described in the chapter.

Chapter 8 introduces the classification problem with the classical approach of linear discriminant analysis which leads to the nonparametric approach in Chapter 9 sometimes called recursive partitioning but because of the fundamental book Classification and Regression Trees by Brieman, Olshen, Friedman and Stone the more common and popular term is tree-based methods.In Chapter 9 Izenman also includes extensions to these methods which include survival trees and Friedman's multivariate adaptive regression splines.Other approaches coming from the disciplines of artificial intelligence and computer science are the subjects of Chapters 10 and 11, neural netowrks and support vector machines respectively.

Chapter 12 covers unsupervised learning through techniques called cluster analysis methods.Chapter 13 covers multidimensional scaling (here color plays a useful role).Chapter 14 is called Committee Machines.This incorporated the great breakthroughs to improving classification algorithms; bootstrap aggregating which Breiman called "bagging" and boosting algorithms of Schapire and Freund in the early 1990s.Also random forests which introducing a randomization component to bagging also due to Brieman is also discussed.

Later chapters include nonlinear dimensionality reduction, exploratory factor analysis and ending up with a multivariate technique called correspondence analysis that got a lot of attention by the french school of statisticians but was largely ignored in the US for many years.

Aside from the many unique and modern topics discussed in this book what really sets it apart is the academic thoroughness from including a large bibliography of over 550 references, with bibliographic notes at the end of each chapter, illustrative and relevant examples expertly placed throughout the chapters, numerous homework exercises starting with Chapter 2 and a list of software tools for implementing the methods where applicable (every chapter from 7 through 17).As a statistician with interest in bootstrap methods I was particularly pleased with the heavy emphasis on the use of the bootstrap where it has been most successful and gratified that in addition to referencing the commonly referenced bootstrap texts Efron and Tibshirani(1993), Davison and Hinkley(1997) and Hall(1992), he also mentions Chernick (1999).Very little was left out on modern methods.The only things I can think of that are not included are the use of influence functions to detect multivariate outliers as described in Gnandesikan's text and the work of Pesarin and his colleagues on multivariate permutation tests.

Know what you're getting
Another reviewer compared this book to Bishop's PRML, and the analogy is a good one.If you liked PRML then you'll love this book, which is similarly 'useful' as a reference from which it would be extremely difficult to learn anything that you didn't already know.If you aren't already extremely sophisticated wrt the material covered here (check the TOC) then a far better approach to learning it would be to cobble together your own curriculum from various academic papers and tutorials and code snippets on these same topics available on the web.

Of course, this is a ridiculous state of affairs, but it's the one in which we find ourselves -- tomes like this one, and Bishop's, of massive and overarching scope, are essentially useless as a means to actually learn material for the first time, and there don't seem to be any decent alternatives.I suppose this shouldn't surprise me -- considering that, for instance, one can find scads of 40 page papers devoted _entirely_ to independent components analysis, what should one expect of a book that attempts to cover that topic in far fewer?Probably it's my own damn fault for not figuring this out.

So if you're superbad at statistical learning, and you want a concise reference to a lot of stuff in one place to jog your memory, this book's solid.If you want to learn about these topics, this book will be an aggravating waste of money, albeit less aggravating than Bishop.

Nice material or PhD students
Good observation of modern approaches for classification and clustering problems. Nice structure of material and nice paper =)

Good book - For Statistics majors
This is not Johnson and Wichern or TW Anderson - Think Bishop (PRML) or Hastie, Tibshirani and Friedman (EoSL). We used this for a course last year and this is a great book -as opposed to Bishop which treats things form a Com. Sci. perspective or HTF which assumes a much higher level. One warning though - don't be turned off by the multivariate notation (Duh... Look at the title, of course), but once you master the early chapter on matrix theory and analysis, everything else is very readable.

nice reference
This book not only covers very wide ranges about ststistical learning but also has very deep discriptions in some topics. This is a good book especially for graduate students. ... Read more

Product Description

A brand new appendix by Oscar Garcia-Prada graces this third edition of a classic work. In developing the tools necessary for the study of complex manifolds, this comprehensive, well-organized treatment presents in its opening chapters a detailed survey of recent progress in four areas: geometry (manifolds with vector bundles), algebraic topology, differential geometry, and partial differential equations. Wells’s superb analysis also gives details of the Hodge-Riemann bilinear relations on Kahler manifolds, Griffiths's period mapping, quadratic transformations, and Kodaira's vanishing and embedding theorems. Oscar Garcia-Prada’s appendix gives an overview of the developments in the field during the decades since the book appeared.

Customer Reviews (8)

tt
this book is vastly better then browder or spivak.it is also more thorough in its discussion of elementary results, (though less thorough in generalities).very helpful for undergraduates.

Excellent textbook on a difficult subject
I took a course in advanced analysis, in which we covered the first few chapters of this book (upto the implicit function theorem). Since then I have been going it alone and have finished integration; on my way to manifolds. An excellent book for a reading course, very lucidly written. Since there are many things which are "obvious" in R, but can really cause difficulties in R^n (path dependence of continuity, for example), the author does a great job of identifying these particular issues (note the entire chapter on change of variables) and pointing out how the difficulties arise.

The one shortcoming of this book is that none of the exercises have any solutions. If the author provided solutions for even 5% of the problems (see, for example, Oksendal's Stochastic Differential Equations), it would have been enormously beneficial for somebody going it alone, like me.

Satisfied customer
Very satisfied with both the service and book. It was used but in very good condition.

Excellent.
I've just finished all but the last half of the last section, which deals with abstract manifolds, and I've done most of the problems in the book. It is important to note that the book only deals with manifolds that are subsets of euclidean n-space.

Anyway, the book is well-written. It demands some maturity and basic linear algebra, analysis and topology. I found only two misprints which are basically of no consequence. Figures abound and are excellent. I've got only two complaints:

(1) The author never mentions that the set of all C^r scalar maps on an open set in R^n is closed under sums, products and quotients. This is used constantly in the latter parts of the book but is never proven. The proof can be found in Spivak's book. The first time this fact is needed is in the proof of the inverse function theorem (det(Df(x)) is a continuous function of x if f is C^r), and also during the construction of a partition of unity. There are more subtle points than this that are left to the reader, but I feel that it should have been proven or given as an exercise if only for the sake of completeness.

(2) The book isn't hard (though it isn't totally easy), but the very last section on abstract manifolds seems harder to read than all the rest of the book, because the author does less to elucidate things here of all places, where more elucidation is needed. He's trying in several pages to generalize results on euclidean submanifolds obtained throughout the whole book to abstract manifolds. I feel that the exposition ought to have been much more thorough here, or much more informal, or that this section should have just been completely omitted.

Nonetheless I feel I'm now ready to take a course in abstract differentiable manifolds. The problems in the book are good, and there are only at most ten or twelve problems in every section, so the reader isn't overburdened as reading the text well and carefully is a task in its own right.

I've profited considerably by completing this book and I highly recommend it.

Fun
Iploughed through this book years ago. Ijust noticed that a couple of reviews were only posted this year.
I thought I would do the same.

This was a great read by the way.

I suspect that everyone who picked up this book at some point was looking fora way to circumvent Spivak's terse exposition. I don't blame them.

..andthenBrowder came out with his analysis text. So with advanced calculus in view, these (more or less)recent publications make the subject even more accessible to undergraduates.

..and nowSpivak doesn't look so hard, all of a sudden.

Munkres presentation is certainly original.Motivating examples are bountiful, and the figures are excellent.

The perfect prequel to Boothby.

Enjoy. ... Read more

Product Description
This book is concerned with statistical analysis on the two special manifolds, the Stiefel manifold and the Grassmann manifold, treated as statistical sample spaces consisting of matrices. The former is represented by the set of m x k matrices whose columns are mutually orthogonal k-variate vectors of unit length, and the latter by the set of m x m orthogonal projection matrices idempotent of rank k. The observations for the special case k=1 are regarded as directed vectors on a unit hypersphere and as axes or lines undirected, respectively. Statistical analysis on these manifolds is required, especially for low dimensions in practical applications, in the earth (or geological) sciences, astronomy, medicine, biology, meteorology, animal behavior and many other fields. The Grassmann manifold is a rather new subject treated as a statistical sample space, and the development of statistical analysis on the manifold must make some contributions to the related sciences. The reader may already know the usual theory of multivariate analysis on the real Euclidean space and intend to deeper or broaden the research area to statistics on special manifolds, which is not treated in general textbooks of multivariate analysis.

The author rather concentrates on the topics to which a considerable amount of personal effort has been devoted. Starting with fundamental material of the special manifolds and some knowledge in multivariate analysis, the book discusses population distributions (especially the matrix Langevin distributions that are used for the most of the statistical analyses in this book), decompositions of the special manifolds, sampling distributions, and statistical inference on the parameters (estimation and tests for hypotheses). Asymptotic theory in sampling distributions and statistical inference is developed for large sample size, for large concentration and for high dimension. Further investigated are Procrustes methods applied on the special manifolds, density estimation, and measurement of orthogonal association.

This book is designed as a reference book for both theoretical and applied statisticians. The book will also be used as a textbook for a graduate course in multivariate analysis. It may be assumed that the reader is familiar with the usual theory of univariate statistics and a thorough background in mathematics, in particular, knowledge of multivariate calculation techniques. To make the book self-contained, a brief review of some of those aspects and related topics is given.

Yasuko Chicuse is Professor of Statistics and Mathematics at Kagawa University, Japan. She earned a Ph.D. in Statistics from Yale University and Sc.D. in Mathematics from Kyushu University, Japan. She is a member of the International Statistical Institute, the Institute of Mathematical Statistics, the American Statistical Association, the Japan Statistical Society, and the Mathematical Society of Japan. She has held visiting research and/or teaching appointments at the CSIRO, the University of Pittsburgh, the University of California at Santa Barbara, York University, McGill University, and the University of St Andrews. ... Read more

Product Description
This book is a collection of 375 completely solved exercises on differentiable manifolds, Lie groups, fibre bundles, and Riemannian manifolds. The exercises go from elementary computations to rather sophisticated tools. It is the first book consisting of completely solved problems on differentiable manifolds, and therefore will be a complement to the books on theory. A 42-page formulary is included which will be useful as an aide-mémoire, especially for teachers and researchers on these topics.

The book includes 50 figures and will be useful to advanced undergraduate and graduate students of mathematics, theoretical physics, and some branches of engineering. ... Read more

Customer Reviews (1)

This book rocks like Newton and Gauss....
Man, this is a very instructional manual that can help just about anyone having problems learning differential geometry. I give it two tangent vectors up. ... Read more

Product Description
The last ten years have seen rapid advances in the understanding of differentiable four-manifolds, not least of which has been the discovery of new 'exotic' manifolds. These results have had far-reaching consequences in geometry, topology, and mathematical physics,and have proven to be a mainspring of current mathematical research. This book provides a lucid and accessible account of the modern study of the geometry of four-manifolds.Consequently, it will be required reading for all those mathematicians and theoretical physicists whose research touches on this topic. The authors present both a thorough treatment of the main lines of these developments in four-manifold topology--notably the definition of new invariants of four-manifolds--and also a wide-ranging treatment of relevant topics from geometry and global analysis.All of the main theorems about Yang-Mills instantons on four-manifolds are proven in detail.On the geometric side, the book contains a new proof of the classification of instantons on the four-sphere, together with an extensive discussion of the differential geometry of holomorphic vector bundles.At the end of the book the different strands of the theory are brought together in the proofs of results which settle long-standing problems in four-manifolds topology and which are close to the frontiers of current research. Co-author Donaldson is the 1994 co-recipient of the prestigious Crafoord Prize. ... Read more

Customer Reviews (1)

An excellent summary of Donaldson theory
This book brings together the brilliant work Donaldson did at Oxford during the early 1980s. The unique properties of 4-manifolds are clearly and concisely written out with concentration on explaining field theorieslike Yang-Mills and gauge theory with a truly firm mathematical foundation,presented in a book for the first time. A great companion for anyresearcher in the field of geometry and topology, or even loop quantumgravity! ... Read more

Product Description
This volume offers an expanded version of lectures given at the Courant Institute on the theory of Sobolev spaces on Riemannian manifolds. "Several surprising phenomena appear when studying Sobolev spaces on manifolds," according to the author. "Questions that are elementary for Euclidean space become challenging and give rise to sophisticated mathematics, where the geometry of the manifold plays a central role."The volume is organized into nine chapters. Chapter 1 offers a brief introduction to differential and Riemannian geometry. Chapter 2 deals with the general theory of Sobolev spaces for compact manifolds. Chapter 3 presents the general theory of Sobolev spaces for complete, noncompact manifolds. Best constants problems for compact manifolds are discussed in Chapters 4 and 5. Chapter 6 presents special types of Sobolev inequalities under constraints. Best constants problems for complete noncompact manifolds are discussed in Chapter 7. Chapter 8 deals with Euclidean-type Sobolev inequalities. And Chapter 9 discusses the influence of symmetries on Sobolev embeddings. An appendix offers brief notes on the case of manifolds with boundaries.This topic is a field undergoing great development at this time. However, several important questions remain open. So a substantial part of the book is devoted to the concept of best constants, which appeared to be crucial for solving limiting cases of some classes of PDEs.The volume is highly self-contained. No familiarity is assumed with differentiable manifolds and Riemannian geometry, making the book accessible to a broad audience of readers, including graduate students and researchers. ... Read more