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This work shows how the concepts of manifold theory can be used to describe the physical world. The concepts of modern differential geometry are presented in this comprehensive study of classical mechanics, field theory, and simple quantum effects. ... Read more

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clear and easily understandable
As an introductory text, I personal think, a reasonable motivation of the subject is the key to success of a book. And that's exactly why it deserve 5 stars, and my 1st time review.

Rich table of contents, poor mathematics
This book has a superb selection of topics; however, if you are interested in a quick but useful introduction to modern mathematics and its physicalapplications, you will realize this is not the book for you. The conceptsare not clear and the math presentation lacks both the geometrical aspectsand rigor which make Nakahara's "Geometry, Topology and Physics"far more understandable. ... Read more

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This second edition of Tensors and Manifolds is based on courses taken by advanced undergraduate and beginning graduate students in mathematics and physics, giving an introduction to the expanse of modern mathematics and its application in modern physics. It aims to fill the gap between the basic courses and the highly technical and specialized courses which both mathematics and physics students require in their advanced training, while simultaneously trying to promote, at an early stage, a better appreciation and understanding of each other's discipline. The book sets forth the basic principles of tensors and manifolds, describing how the mathematics underlies elegant geometrical models of classical mechanics, relativity and elementary particle physics. The existing material from the first edition has been reworked and extended in some sections to provide extra clarity, with additional problems. Four new chapters on Lie groups and fibre bundles have been included, leading to an exposition of gauge theory and the standard model of elementary particle physics. Mathematical rigour combined with an informal style makes this a very accessible book and will provide the reader with an enjoyable panorama of interesting mathematics and physics. ... Read more

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This book is an introduction to the remarkable work of Vaughan Jones and Victor Vassiliev on knot and link invariants and its recent modifications and generalizations, including a mathematical treatment of Jones-Witten invariants. It emphasizes the geometric aspects of the theory and treats topics such as braids, homeomorphisms of surfaces, surgery of 3-manifolds (Kirby calculus), and branched coverings. This attractive geometric material, interesting in itself yet not previously gathered in book form, constitutes the basis of the last two chapters, where the Jones-Witten invariants are constructed via the rigorous skein algebra approach (mainly due to the Saint Petersburg school).

Unlike several recent monographs, where all of these invariants are introduced by using the sophisticated abstract algebra of quantum groups and representation theory, the mathematical prerequisites are minimal in this book. Numerous figures and problems make it suitable as a course text and for self-study. ... Read more

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This volume serves as an introduction to the Kodaira-Spencer theory of deformations of complex structures. Based on notes taken by James Morrow from lectures given by Kunihiko Kodaira at Stanford University in 1965-1966, the book gives the original proof of the Kodaira embedding theorem, showing that the restricted class of Kähler manifolds called Hodge manifolds is algebraic. Included are the semicontinuity theorems and the local completeness theorem of Kuranishi. Readers are assumed to know some algebraic topology. Complete references are given for the results that are used from elliptic partial differential equations. The book is suitable for graduate students and researchers interested in abstract complex manifolds. ... Read more

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This book introduces the concepts of tensor algebras and differentiable manifolds to the intermediate-level student. It describes analytical and geometrical structures built on these basic concepts. Those structures -- which include differential forms and their integration, flows, Lie derivatives, distributions and their integrability conditions, connections, and pseudo-Riemannian and symplectic manifolds -- are then applied to the description of the fundamental ideas and Hamiltonian and Lagrangian mechanics, and special and general relativity. This book is designed to be accessible to the mathematics or physics student with a good standard undergraduate background, who is interested in obtaining a broader perspective of the rich interplay of mathematics and physics before deciding on a specialty. ... Read more

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Tensors and Manifolds: with Applications to ...
Reviewer: Gonzalo Torroba from La Pampa, ArgentinaIt's and excellent book where you can see all the power and ellegancy that tensors have in physics. In "Tensors and manifolds..." you study mathematics in aclear and understandable way, knowing it's purpose. Some little points: Ithink it does not have enough applications about vector calculus (Frenetformulas, scalar potentials, Maxwell equations...).I would have alsopreffered it to contain more about general tensor theory: pseudotensors,curvilineal coordinates, and also about Riemann spaces. ... Read more

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Topology of Surfaces, Knots, and Manifolds offers an intuition-based and example-driven approach to the basic ideas and problems involving manifolds, particularly one- and two-dimensional manifolds. A blend of examples and exercises leads the reader to anticipate general definitions and theorems concerning curves, surfaces, knots, and links--the objects of interest in the appealing set of mathematical ideas known as "rubber sheet geometry." The result is a book that provides solid coverage of the mathematics underlying these topics. ... Read more

Customer Reviews (3)

Fun, but not very substantive.
This is a relatively fun romp through some very interesting concepts, but it lacks rigor.The book could have been much stronger if the author had simply developed some of the basic concepts (compactness, connectedness, homeomorphisms, homotopy, etc) rather than do a little hand-waving around a nice illustration.As it stands, this book is only 140 pages long, and does not develop any of its topics (manifolds, surfaces, graphs, knots) adequately.This book is far too weak to serve as a good text.Kinsey's TOPOLOGY OF SURFACES is much stronger, and costs less.Or look as something like Gamelin's INTRO TO TOPOLOGY.Or even Schaum's outline GENERAL TOPOLOGY, which deals with the basics, but is highly readable and rigorous.

Very Misleading Title, Quite Thin, No Rigor, and Overpriced
This book is subtitled "A First Undergraduate Course" but is certainly below undergraduate level.A high school student could easily follow this--which might be a good thing in certain cases--but the rigor is lacking.In fact, there is barely a hint of any rigor whatsoever.It is mostly intuitive arguments and the author often says things like "but we won't bother worrying about mathematical technicalities".It does get you to be able to visualize certain things well, but the visualization techniques can be found in other books also.The book is very thin and a quick read--hardly worth the money they are trying to get for it.If you're really at the undergraduate level and want to learn some topology, try something like Mendelson's "Introduction to Topology" by Dover or one of the excellent topology books in the series "Undergraduate Texts in Mathematics" by Springer.Munkres is also a classic.If you're not an undergraduate in a math related field and just want to know about the ideas behind topology or perhaps see some visualization techniques, try something like "The Shape of Space" by Weeks.Overall I was very disappointed with this text.If you could purchase this book for under $20 it might be worth it, but even then I think the other books I quoted are better in both price and substance.

No math library is complete without this book
This book presents the topology of surfaces, manifolds and knots in a manner that is reachable for undergraduate students with only a knowledge of calculus.Some linear algebra might be helpful.The text is written in a style that is easy to follow and there are superfluous examples.The exercises in the text are well thought out and are not extremely difficult.The exercises complement the text very well.The text makes clear a lot of difficult concepts such as isotopic surfaces as opposed to homeomorphic surfaces.I particularly enjoyed the manner in which the topology of knots was explained.After reading this text, the reader should be able to better visualize the projective plain and even the Klein bottle as it exists in 4-dimensional space.I have not read a text on topology that I enjoyed reading as much since Munkres.This text is a must have for any topologist. ... Read more

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HPBOOKS HP1052 HOLLEY CARBS, MANIFOLDS, FI ... Read more

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How to tune a holley
This book tells you how Holley metering systems work, has a graphical carb size selector on page 76, gives manifold characteristics, holley carb tuning info, has exploded diagrams, and a disassembly chapter with pictures. This would be a decent rebuild guide.

"fuel" for thought
Well, I bought this book mainly because I was installing a Holley analog Pro-Jection TBI fuel injection system. It covers many holley carbs from 2 barrels, 4150's, 4160's, and 2 and 4 barrel pro-jection. I was surprised slightly that they didn't cover the "economaster-4360" but then again, this carb has been out of production for over a decade. It would have been a PERFECT book if it covered the 4360 but overall. This is a very worthwhile book, whether you are looking to learn about Holley carburetors or fuel injection. ... Read more

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This second edition, divided into fourteen chapters, presents a comprehensive treatment of contact and symplectic manifolds from the Riemannian point of view. The monograph examines the basic ideas in detail and provides many illustrative examples for the reader.

Riemannian Geometry of Contact and Symplectic Manifolds, Second Edition provides new material in most chapters, but a particular emphasis remains on contact manifolds. New principal topics include a complex geodesic flow and the accompanying geometry of the projectivized holomorphic tangent bundle and a complex version of the special directions discussed in Chapter 11 for the real case. Both of these topics make use of Étienne Ghys's attractive notion of a holomorphic Anosov flow.

Researchers, mathematicians, and graduate students in contact and symplectic manifold theory and in Riemannian geometry will benefit from this work. A basic course in Riemannian geometry is a prerequisite.

Reviews from the First Edition:

"The book . . . can be used either as an introduction to the subject or as a reference for students and researchers . . . [it] gives a clear and complete account of the main ideas . . . and studies a vast amount of related subjects such as integral sub-manifolds, symplectic structure of tangent bundles, curvature of contact metric manifolds and curvature functionals on spaces of associated metrics."   —Mathematical Reviews

"…this is a pleasant and useful book and all geometers will profit [from] reading it. They can use it for advanced courses, for thesis topics as well as for references. Beginners will find in it an attractive [table of] contents and useful ideas for pursuing their studies."   —Memoriile Sectiilor Stiintifice

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This volume is a reprint with few corrections of the original work published in 1964. Starting with the notion of differential manifolds, the first six chapters lay a foundation for the study of Riemannian manifolds through specializing the theory of connections on principle bundles and affine connections. The geometry of Riemannian manifolds is emphasized, as opposed to global analysis, so that the theorems of Hopf-Rinow, Hadamard-Cartan, and Cartan's local isometry theorem are included, but no elliptic operator theory. Isometric immersions are treated elegantly and from a global viewpoint. In the final chapter are the more complicated estimates on which much of the research in Riemannian geometry is based: the Morse index theorem, Synge's theorems on closed geodesics, Rauch's comparison theorem, and the original proof of the Bishop volume-comparison theorem (with Myer's Theorem as a corollary).

The first edition of this book was the origin of a modern treatment of global Riemannian geometry, using the carefully conceived notation that has withstood the test of time. The primary source material for the book were the papers and course notes of brilliant geometers, including É. Cartan, C. Ehresmann, I. M. Singer, and W. Ambrose. It is tightly organized, uniformly very precise, and amazingly comprehensive for its length. ... Read more

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A Unique Classic
Differential geometry is one of the most highly developed subjects in all of mathematics.The literature is daunting, both in volume and complexity.The serious student will soon learn that there simply is no single "perfect book" on the subject from which one can learn everything one needs to know.This is doubly true for the student who wants to learn about both Riemannian manifolds and Semi-Riemannian geometry, the language of Einstein's theory of general relativity.

The book by Bishop and Crittenden, long out of print and difficult to find before this recent re-printing emerged, contains a wealth of important and fundamental insights that are simply not to be found in any other differential geometry text.I will describe only one example in detail;many other examples of a similar nature could be cited.

As one studies differential geometry, one quickly learns that there are uncountably many connexions on a typical manifold (M,g).However, most books quickly restrict their attention to the Levi-Civita connexion, the unique connexion that is (1) metrically compatible, and (2) has torsion zero.

While metric compatibility is fairly easy to understand, the notion of torsion zero is far more elusive.Do a quick internet search and you will find scores of hapless students who are begging for help in understanding the GEOMETRIC content of the torsion tensor. Students of general relativity quickly learn that the mathematical expression of Einstein's Equivalence Principle will not hold unless the connexion has zero torsion, and that is sufficient to motivate the condition in GR;however, this still does not explain the tensor's geometric content.

Bishop and Crittenden give a visual interpretation of torsion in terms of geodesic quadrilaterals (see page 97) that will appeal to anyone who is searching for geometric intuition.I have over 150 differential geometry books in my personal library, and Bishop and Crittenden is the only one to provide this intuitive, geometric understanding of the torsion tensor.Richard Bishop continued this trend in his later book, co-authored withSam Goldberg, where he gives a similar geometric interpretation of the Lie bracket.

If you are a devoted student of geometry, then I suggest you add Bishop and Crittenden to your library, along with Spivak (5 volumes), Kobayashi and Nomizu (2 volumes), Chern, etc., etc.Each of these references contains unique insights not to be found in any of the others.You may only need to refer to this book a few times, but the insights gained will be well worth the meager purchase price.

excellent but not a first semester textbook
I recommend this as a supplement for students who have already learned Riemannian Geometry or at least Differential Geometry elsewhere for 1-2 semesters.It is excellent for fibre bundles and viewing a metric as an SO(n) bundle over a manifold.It is also an excellent reference for Bishop's Volume Comparison Theorem which has since been adapted by Gromov in his book "Metric Structures for Riemannian and Non-Riemannian Spaces" and has led to a fundamental change in the study of manifolds with Ricci curvature bounds.

Prerequisite books I read as a grad student were Spivak's Differential Geometry and DoCarmo's "Riemannian Geometry".

Spectacular geometric insight into differential geometry
As a differential geometer for the past 30 years, I own 8 introductions to the field, and I have perused a half-dozen others. Bishop & Crittenden's "Geometry of Manifolds" is unparallelled for imparting a strong geometric intuition about Lie derivatives and connections on fiber bundles, which is the key to understanding this field, plus general relativity, Einstein-Cartan theory, and gauge theories in physics. Bishop & Crittenden taught me to see pictures for the main constructions and theorems, though I admit I had to work hard to build my intuition. It is particularly strong in establishing a 1-1 correspondence between geometric images of horizontal planes in fiber bundles and the language of differential forms in fiber bundles. I first read the classic Chevalley's "Lie Groups" which is very strong on geometric intuition about manifolds and Lie groups. I also relied on Kobayashi & Numizu's "Foundations of Differential Geometry" I & II because ity has better algebraic and computational treatment in some areas, and it cover topics not in "Geometry of Manifolds."

If you are really serious about differential geometry, if you prefer to think in terms of geometric visualization when possible and use algebra afterwards, then this book is the greatest -- nothing I have seen comes close. If you prefer to think mainly with abstract algebra, then you might prefer Kobayashi and Nomizu.

Richard J. Petti ... Read more

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The purpose of this book is to provide core material in nonlinear analysis for mathematicians, physicists, engineers, and mathematical biologists. The main goal is to provide a working knowledge of manifolds, dynamical systems, tensors, and differential forms. Some applications to Hamiltonian mechanics, fluid mechanics, electromagnetism, plasma dynamics and control theory are given using both invariant and index notation. The prerequisites required are solid undergraduate courses in linear algebra and advanced calculus. ... Read more

Customer Reviews (4)

A complete book by very erudite authors
I actually read this entire book--it is quite long and dense. Actually I took the course from the author Jerry Marsden at Caltech and Tutor (Jerry's friend and co-author) gave a guest lecture while visiting. We flew through the entire thing and ch 9 on lie groups of his mechanics and symmetry text in a short 10 weeks! My background in math was relatively weak when taking the course so it was a little hard to keep up; i.e. I came from an engineering background. Anyway, it is probably the most complete/diverse text I've come across on the subject. Of course, it's actually more of a monograph than a text. Since I've read the whole thing, I have to admit there are "several" typos. But as it is that most people can't even write a damn email without a typo or two, the book really does a good job considering it is 800 pages of mostly dense mathematical rigor. I imagine that if I wrote 800 pages of mathematical symbols in latex, that I might forget a tilde or put something as subscript that should have been superscript here or there! None of these errors really matter too much-they should not hinder one's understanding. All and all I think that this book is a great ref, although I've never seen the index, if one exists. For the beginner, also check out Boothby's book, which covers a lot of the same material but tones it down a bit.

A Unique Reference
Students of mathematical physics in general, and general relativity in particular, face a formidable challenge in attempting to find coherent, readable references on manifold theory and tensor analysis.I think it fair to say that for every well-written work on the subject, there are ten that do more damage than good.Very few texts can claim to (1) be clear enough to assist the person who is studying alone, (2)offer valuable PHYSICAL insight into the subject, and (3) pass the standards of rigor that mathematicians would impose.Abraham, Marsden, and Ratiu manage to accomplish all three of these goals in this profoundly useful text.I studied from the first edition and I have taught from the second.The two chapters on differential forms, Hodge star duality, integration on manifolds, and the generalized Stokes' Theorem alone are worth the price of the entire book.I am unaware of any other reference which which treats differential forms with the same combination of clarity, physical motivation, and mathematical rigor.The concluding chapter on applications offers one of the clearest introductions to the relativistic form of Maxwell's equations to be found in any text.For students of physics who want to see the mathematics "done right," one would be hard pressed to do better than Abraham, Marsden, and Ratiu.

mixed bag: many virtues but many weaknesses
I took a course taught by the 3rd author (Tudor Ratiu) at UCSC using this book; I found both good and bad in it.Much of the bad for me was overcome by the inspiring and energetic presentation by one of the authors.One mayview this book as basically a detailed elaboration of the"preliminary" chapters of the book "Foundations ofMechanics" by the 1st 2 authors. The strengths of this book are (a)the treatment which is general enough to include infinite-dimensionalmanifolds and not just the finite-dimensional case (most books just talkabout the finite-dim'l case) and (b) the attempt to cover all theorems"full strength" (in the greatest generality obtaining thestrongest conclusions from the weakest hypotheses). Neither of these (notcounting the many typos) recommends this as a first or even second text forstudents, but it's hard to find any other books that treat the material atthe same level of generality and precision, which is a must if attempting"hard" global analysis in areas such as fluid mechanics (from ageometric point of view).Correction of the many typos could make this anindispensable reference book for those requiring the techniques discussed.More discussion of finite-dimensional examples before jumping toinfinite-dimensional ones (e.g. discussing finite-dimensional Grassmanniansbefore jumping to the infinite-dimensional Banach manifold version) couldmake this into a tolerable text.

As it is, it's problematic, aggravating,and not for the faint of heart, but not without its virtues.

Possiblealternatives for the infinite-dimensional point of view are Lang'smanifolds book or some volume of the expensive multi-volume treatise onanalysis by Dieudonne.

Poorly writen, filled with errors, very long, poorly indexed
We used this book in a graduate course at UCLA. The professor had to hand out a list of all the errors we encountered, and it was about ten pages typewritten. The professor, Geoff Mess, wrote at the top of this list thatmany of the students had complained about this book, and that it was adisappointment to him as well.I often found myself scanning hundreds ofpages in search of what should have been contained in their sparse index.The book is unnecessarily long and wordy for the matter covered. In theintroduction, the authors mention that they invite comments from thereaders. It seems that they depend on their readers to correct theircopious errors and their poor writing. ... Read more

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Provides an introduction to the geometry of foliations and covers local and global results on Riemannian manifolds with foliations, including rigidity, splitting, and integral formulas. Gives a survey of submanifolds with generators. DLC: Riemannian manifolds. ... Read more

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For those working in singularity theory or other areas of complex geometry, this volume will open the door to the study of Frobenius manifolds. In the first part Hertling explains the theory of manifolds with a multiplication on the tangent bundle. He then presents a simplified explanation of the role of Frobenius manifolds in singularity theory along with all the necessary tools and several applications. Readers will benefit from this careful and sound study of the fundamental structures and results in this exciting branch of mathematics. ... Read more

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From the reviews of the first edition: "This book exposes the beautiful confluence of deep techniques and ideas from mathematical physics and the topological study of the differentiable structure of compact four-dimensional manifolds, compact spaces locally modeled on the world in which we live and operate... The book is filled with insightful remarks, proofs, and contributions that have never before appeared in print. For anyone attempting to understand the work of Donaldson and the applications of gauge theories to four-dimensional topology, the book is a must." #Science#1 "I would strongly advise the graduate student or working mathematician who wishes to learn the analytic aspects of this subject to begin with Freed and Uhlenbeck's book." #Bulletin of the American Mathematical Society#2 ... Read more

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the best introduction to Donaldson's gauge theory, even if it is outdated
Gauge theory as a field for study by mathematicians, and in particular, topologists, was initiated by Donaldson's proof, which is the content of this book, that the only simply connected smooth closed 4-manifolds with definite intersection form are connected sums of complex projective planes (with the factors having either orientation). This was the first major advance in 4-manifold smooth topology beyond Rohlin's theorem on the signature of a smooth spin 4-manifold and the Kirby-Siebenmann obstruction. Donaldson's theorem was followed by many other related results, in particular, the discovery of the Donaldson polynomial invariants and various gluing and vanishing theorems, none of which appear in this early book, so this work is not suitable as a comprehensive reference for the theory. But it is still essential for a student wishing to learn gauge theory because of the extensive coverage of the requisite analysis, topology, and geometry that one needs to work in this field (or even read any later works). The authors endeavor to explain intuitively the motivation behind the different steps in this long proof (it takes more than 100 pages to prove the theorem), with some effective figures included as well.

Donaldson's proof is based on studying the moduli space of certain (anti-self-dual) solutions (mod gauge equivalences) of the Yang-Mills equations, the most fundamental equation in particle physics, which includes the vacuum Maxwell equations of electromagnetics, as well as those of QED and QCD, as special cases. For the SU(2) case over a smooth simply connected compact 4-manifold, the moduli space can be shown to be nonempty, orientable, smooth except for a finite number of points, near which the moduli space is a cone on a projective space, and having one noncompact end that is diffeomorphic to a collar on the original 4-manifold. This establishes a cobordism between the original manifold and a disjoint union of copies of CP^2, which proves the theorem. The various chapters of the book prove each of the aforementioned properties of the moduli spaces, mainly through tedious technical analytic results. Along the way the necessary material concerning bundles, connections, Sobolev spaces, Fredholm operators, index theory, classifying spaces, Bochner-Weitzenboeck formulas, etc., is presented, making this the most accessible gauge-theoretic work for (2nd-year) graduate students.

The second author, Karen Uhlenbeck, was one of the pioneers in the field, deriving several critical analytic results about sequences of solutions to the YM equations that are ubiquitous in gauge theory proofs; the book includes her proof of the famed removable singularities theorem as an appendix. There's also an early section on the construction of fake R^4s, one of the strangest results of 4-manifold topology (although I think Lawson's Theory of Gauge Fields in Four Dimensions does a better job of explaining this, and orientability as well). It is preferable to read the second edition, which includes a brief introduction that sumarizes advances in the field between 1984 and 1991, in addition to having been re-typeset.

A demerit of the book is that it is filled with mathematical typos and many of the technical proofs require some modification in the choice of constants owing to some errors in the inequalities. However, the theorems themselves remain true, as the reader can verify with some effort.

This book will provide the reader the tools to proceed to Donaldson and Kronheimer's The Geometry of Four-Manifolds and Morgan's Gauge Theory and the Topology of Four-Manifolds and The Seiberg-Witten Equations and Applications to the Topology of Smooth Four-Manifolds. ... Read more

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This text on analysis on Riemannian manifolds is a thorough introduction to topics covered in advanced research monographs on Atiyah-Singer index theory. The main theme is the study of heat flow associated to the Laplacians on differential forms. This provides a unified treatment of Hodge theory and the supersymmetric proof of the Chern-Gauss-Bonnet theorem. In particular, there is a careful treatment of the heat kernel for the Laplacian on functions. The author develops the Atiyah-Singer index theorem and its applications (without complete proofs) via the heat equation method. Rosenberg also treats zeta functions for Laplacians and analytic torsion, and lays out the recently uncovered relation between index theory and analytic torsion. The text is aimed at students who have had a first course in differentiable manifolds, and the author develops the Riemannian geometry used from the beginning. There are over 100 exercises with hints. ... Read more

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Fascinating book - I couldn't put it down
Steve Rosenberg's writing style is fascinating.I was up until 3 in the morning finishing this book.

Pithy plot, great characters
A real page turner!It was unbelievable how all the pieces fit together at the end.Rosenberg is a genius.

Good introduction but not detailed enough
Although a good introduction is given certains ideas are not really worked out in detail (proofs,...) the level may be put a little bit higher ... Read more

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From the reviews:

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A Japanese mathematician
Kunihiko Kodaira is one of the greatest japanese mathematicians.

He is one of the last students of Teiji Takagi who is famous
for class field theory.

Kunihiko Kodaira died at Kohu, Yamanashi prefecture, Japan in 1997.

He also wrote essays.

Superb
Of importance to applications such as superstring theories in high-energy physics, the theory of complex manifolds and the deformation of complex structures are explained in great detail in this book by one of the major contributors to the subject. One of the valuable features of the book that is actually rare in more recent books on mathematics is that the author tries (and succeeds) to give motivation for the subject. This feature is actually quite common in older books on mathematics, for with few exceptions writers at that time believed that a proper understanding of mathematics can only come with explanations that are given outside the deductive structures that are created in the process of doing mathematics. These explanations frequently involve the use of diagrams, pictures, intuitive arguments, and historical analogies, and so are not held to be rigorous from a mathematical standpoint. They are however extremely valuable to students of mathematics and those who are interested in applying it, like physicists and engineers. There seems to be an inverse relationship between rigor and understanding of mathematics, and given the emphasis on the former in modern works of mathematics, one can expect students to have more trouble learning a particular branch of mathematics than those students of a few decades ago.

Luckily though the author of this book has given the reader valuable insights into the nature of complex manifolds and what is means to deform a complex structure. Complex manifolds are different from real manifolds due to the notion of holomorphicity, but are similar in the sense that they are constructed from domains that are "glued together". In complex manifolds, the "glue" is provided by biholomorphic maps between the domains, the latter of which are open sets called `polydisks'. A `deformation' of the complex manifold is then considered to be a glueing of the same polydisks but via a different identification. For an n-dimensional complex manifold, the maps could thus be dependent on say m parameters, which are labeled as "t" by the author. This dependence on t would result in a differentiable family of complex manifolds. One thus expects the complex manifold to be dependent on t, but the author discusses a counterexample that indicates that one must not be cavalier about this approach.

The definition that is arrived at involves letting t be an element of a domain B in m-dimensional Euclidean space, and considering a collection of compact complex n-dimensional manifolds that depends on t. This collection will be a `differentiable family' if: 1. There exists a differentiable manifold M and a C-infinity map W from M onto B such that the rank of the Jacobian matrix of W is equal to m at every point of M. 2. M(t), the inverse image of t under W is a compact connected subset of M, and in fact is equal to a member of the collection. 3. M has a locally finite open covering along with smooth coordinate functions on the covering that have non-empty intersection with each member of the covering. Beginning with an initial element of B, each member of the inverse image of t under W is viewed as a deformation of the initial member. The crucial point made by the author is that the restricting the domain of the parameter t to a sufficiently small interval allows the representation of the member M(t) as a union of polydisks that are independent of t. Therefore only the coordinate transformations depend on t, and thus only the way of glueing the polydisks depends on t.

To show that these constructions are meaningful, namely that the complex structure of M(t) actually depends on t, the author studies the case of m = 1. In the process he constructs the infinitesimal deformation of M(t), and interprets it as the derivative of the complex structure of M(t) with respect to t. He also shows that the infinitesimal deformation does not depend on the choice of systems of local coordinates, and that the infinitesimal deformation vanishes when M(t) does not vary with t. The author then defines, using a notion of equivalence between two differentiable families, a differentiable family (M, B, W) to be `trivial' if it is equivalent to a product (M x B, B, P). Restricting this triviality to a subdomain gives a notion of `local triviality', which implies immediately that each M(t) will be biholomorphically equivalent to a fixed M. He then shows that if the infinitesimal deformation vanishes then the differentiable family is locally trivial. A more substantial statement of this result is encapsulated in the Frolicher-Nijenhuis theorem, which follows from the results that the author proves in the book. These results involve the theory of strongly elliptic differential operators and considerations of the first cohomology group of M(t) with coefficients in the sheaf of germs of holomorphic vector fields over M(t).

The case of a complex analytic family of compact complex manifolds entails that B will be domain in complex n-space. The author shows that a complex analytic family will be trivial if it is trivial as a differentiable family. As expected, because of the nature of analyticity learned from the theory of complex variables, the proof of these results involves the theory of harmonic differential forms. The author gives these proofs in detail in the book. He also considers the question whether if given an element b of the first cohomology group of a compact complex manifold Mwith coefficients in the sheaf of germs of holomorphic vector fields over M, one can find a complex analytic family that takes M as its initial element and the derivative equal to b. This question, as expected, involves the use of obstruction theory, which the author develops in great detail. In these considerations, the reader will see the and origin and role of the moduli of complex structures. These are essentially the number of parameters m, as long as the complex analytic family is `complete.' ... Read more

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Mathematics text for undergraduate students having already studied single-variable calculus. Includes index, glossary, and exercises. ... Read more

Customer Reviews (2)

nil
A very thorough and rigorous treatment of multivariable calculus and linear algebra, though the part on volume as a multiple integral has a little room for expansion. Very adequate examples to illustrate every concept. Nice book overall.

A Very Challenging Combination of Linear Algebra and Multivariate Calculus
At the University of Georgia this book is used for a two semester honors course that combines linear algebra and multivariate calculus (MATH 3500 and MATH 3510) and is generally taught by Dr. Shifrin himself.The idea is to show students how the two subjects are fundamentally related while at the same time giving rigorous proofs of the underpinnings of vector calculus and linear algebra.I would say that about 80% of the work in this text is proof related, so if you're one who normally struggles with theory and proofs this is definitely not the book for you.

While Dr. Shifrin can be a bit dry with his explanations at times I can guarantee you that students will learn a lot by putting forth the effort.Again, for those of you who are not familiar with Shifrin's books or style of writing I would not recommend this book to any beginner of linear algebra and/or multivariate calculus.This book is best served for students who feel comfortable proving theorems and being able to justify their work.Nonetheless, if you're up for a good challenge I would highly recommend this book because the exercises will definitely make you think.

In conclusion, I think one of the most rewarding aspects of this book is the appreciation of linear algebra/vector calculus that one will have after having labored through the chapters.Oftentimes students take linear algebra and leave the class wondering what the point of it all was, which actually defeats the purpose of taking the class in the first place.Well, with this book those questions will be taken care of because students are exposed to the practicality of linear algebra by showing how it is related to calculus and how it is useful in and of itself.If you're looking for more than just mere formulas/generalizations to memorize to get you through the class (or if you're just curious about the subject) then by all means pick up this book because I guarantee you will not be disappointed. ... Read more

Product Description

The second volume of Shafarevich's introductory book on algebraic geometry focuses on schemes, complex algebraic varieties and complex manifolds. As with first volume the author has revised the text and added new material. Although the material is more advanced than in Volume 1 the algebraic apparatus is kept to a minimum making the book accessible to non-specialists. It can be read independently of the first volume and is suitable for beginning graduate students.

Customer Reviews (2)

Are you looking for literary criticism? It's a freaking math book!
The first book in this two volume introduction to algebraic geometry was used as the primary textbook for my algebraic geometry class.It was amazing.Easily the most readable (oh, Hartshorne why are you so heartless?) of all the algebraic algebraic geometry I own (which is quite a few).I finally managed to secure the second volume after several fruitless searches and order cancellations (Amazon didn't have it in stock for months).Though I'm only starting to get into this one and am basing this recommendation mostly on my experience with the first volume, I still think that Shafarevich writes the most accessible introduction to schemes that I've ever read.

Was the book modified without modifying the index?
Shafarevich explains mathematics well, but I find the index for this book extremely frustrating (I bought my copy in the 2005 Springer sale).I'm almost convinced that they forgot to modify the index after making modifications to the first edition. ... Read more