Book Description

Dirac operators play an important role in several domains of mathematics and physics, for example: index theory, elliptic pseudodifferential operators, electromagnetism, particle physics, and the representation theory of Lie groups. In this essentially self-contained work, the basic ideas underlying the concept of Dirac operators are explored. Starting with Clifford algebras and the fundamentals of differential geometry, the text focuses on two main properties, namely, conformal invariance, which determines the local behavior of the operator, and the unique continuation property dominating its global behavior. Spin groups and spinor bundles are covered, as well as the relations with their classical counterparts, orthogonal groups and Clifford bundles. The chapters on Clifford algebras and the fundamentals of differential geometry can be used as an introduction to the above topics, and are suitable for senior undergraduate and graduate students. The other chapters are also accessible at this level so that this text requires very little previous knowledge of the domains covered. The reader will benefit, however, from some knowledge of complex analysis, which gives the simplest example of a Dirac operator. More advanced readers---mathematical physicists, physicists and mathematicians from diverse areas---will appreciate the fresh approach to the theory as well as the new results on boundary value theory. ... Read more

Book Description

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

Gerard Walschap is Professor of Mathematics at the University of Oklahoma where he developed this book for a series of graduate courses he has taught over the past few years. ... Read more

Book Description

The object of this book is to introduce the reader to some of the most important techniques of modern global geometry. In writing it we had in mind the beginning graduate student willing to specialize in this very challenging field of mathematics. The necessary prerequisite is a good knowledge of the calculus with several variables, linear algebra and some elementary point-set topology.

We tried to address several issues. 1. The Language; 2. The Problems; 3. The Methods; 4. The Answers.

Historically, the problems came first, then came the methods and the language while the answers came last. The space constraints forced us to change this order and we had to painfully restrict our selection of topics to be covered. This process always involves a loss of intuition and we tried to balance this by offering as many examples and pictures as often as possible. We test most of our results and techniques on two basic classes examples: surfaces (which can be easily visualized) and Lie groups (which can be elegantly algebraized). When possible we present several facets of the same issue.

We believe that a good familiarity with the formalism of differential geometry is absolutely necessary in understanding and solving concrete problems and this is why we presented it in some detail. Every new concept is supported by concrete examples interesting not only from an academic point of view.

Our interest is mainly in global questions and in particular the interdependence geometry/topology, local/global.

We had to develop many algebraico-topological techniques in the special context of smooth manifolds. We spent a big portion of this book discussing the DeRham cohomology and its ramifications: Poincare duality, intersection theory, degree theory, Thom isomorphism, characteristic classes, Gauss-Bonnet etc.We tried to calculate the cohomology groups of as many as possible concrete examples and we had to do this without relying on the powerful apparatus of homotopy theory (CW-complexes etc.). Some of the proofs are not the most direct ones but the means are sometimes more interesting than the ends. For example in computing the cohomology of complex grassmannians we returned to classical invariant theory and used some brilliant but unadvertised old ideas.

In the last part of the book we discuss elliptic partial differential equations. This requires a familiarity with functional analysis. We painstakingly described the proofs of elliptic Lp and Holder estimates (assuming some deep results of harmonic analysis) for arbitrary elliptic operators with smooth coefficients. It is not a "light meal" but the ideas are useful in a large number of instances. We present a few applications of these techniques (Hodge theory, uniformization theorem). We conclude with a close look to a very important class of elliptic operators namely the Dirac operators. We discuss their algebraic structure in some detail, Weizenbock formulae and many concrete examples. ... Read more

Reviews (1)

Liviu is a genius
I have Liviu as a professor, and the man is a genius. You should buy his book. Straight up. ... Read more

Book Description

Symplectic geometry has its origins as a geometric language for classical mechanics. But it has recently exploded into an independent field interconnected with many other areas of mathematics and physics. The goal of the IAS/Park City Mathematics Institute Graduate Summer School on Symplectic Geometry and Topology was to give an intensive introduction to these exciting areas of current research. Included in this proceedings are lecture notes from the following courses: Introduction to Symplectic Topology by D. McDuff; Holomorphic Curves and Dynamics in Dimension Three by H. Hofer; An Introduction to the Seiberg-Witten Equations on Symplectic Manifolds by C. Taubes; Lectures on Floer Homology by D. Salamon; A Tutorial on Quantum Cohomology by A. Givental; Euler Characteristics and Lagrangian Intersections by R. MacPherson; Hamiltonian Group Actions and Symplectic Reduction by L. Jeffrey; and Mechanics: Symmetry and Dynamics by J. Marsden. ... Read more

Reviews (4)

A good introduction but not the best
One thing i like about this book is the way Munkres presents the counterexamples : why theorem 5.11 wont work if we ease one statement from the hypothesis. Also, the material is accessible and the exercises hard -- both of which, IMHO, are important benchmark for a good math text.

However, compared to his classic textbook of topology, Munkres did not perform as well in connecting with the readers. The text is very hard to read, and is not suitable for self study. This is useful only as a class text, or as a reference for those who already knew (or passed) the subject.

A masterpiece yet accessible on this topic
This book covers a natural extention to my course on analysis in R^n--only content similar to first one sixth of the book got treated at the end of the course. Having read first half (just before manifold) in a continuous fashion (span of nearly a week for 4 hours-ish p.d.), I find this one exceptionally clearly-written, (unlike some point in Spivak's Calculus on Manifold), and in content it serves as a detailed amplification on Spivak's (Sp seems to try to keep the proofs elegant and concise more than possible, making a couple of important theorems render indigestible).

Other noticeable features are:

1) Mistake-free.

2) Proofs are truncated into stages with explicit objectives in each, making them well-structured on paper and easy to recall in future, and in this way techniques in proofs become highlighted into some elementary theorems (to get most job done) so that the scope of applications are much widened.

3) Motivations scattered throughout the book for integrity.

4) Examples given illustrate as counterexample of how theorem fails with some condition changed or missing.

5) The level of presentation is uniform throughout the book: strictly speaking, only a good single-variable analysis course (Rudin will do, and also helpful to refer to the overlapping topics) and some motivation are needed, all essential concepts of linear algebra, topology are introduced afresh and uniquely and in the favorable context: either indispensible in later proofs (can act as a practice of it) or results proven motivate its introduction and properties, though some knowledge beforehand can help you to appreciate more, and focus on mainbody.

6) Each proof is not necessarily the shortest in methods, you may say, but looks most natural and appropriate at this level. Actually, most time it's quite concise whilst, in main theorems, all details are laid out without undue omission. (In contrast, some authors waffle lavishly between substance, but say bare minimum (sometimes unjustified) when it comes to proofs.) Length is also due to partition of proof into stages, which is way clearer in mind than a gluster of dense but appearingly short arguments. And richness and details of proofs themselves are good for getting hang of techniques.

All in all, Munkres is clearly a master, while reading it, you just feel it cannot get any better. Clarity, style, and organisation put the book far above its peers, and an undeniably outstanding first course in multivariable analysis and manifold alike.

Although exam-irrelevant, I will surely continue the journey of reading it, in a belief that it'll serve as a solid step-stone to embark on diff geometry or GR with ease, which is my original purpose. hope you can share my enjoyment.

Extreme clarity, elucidative proofs
This book is one of the most well written mathematics books I've ever read. I found it at a booksale at a local college bookstore, and I sat reading it for 15 minutes in the store. I found the subject matter enthrauling, and I would recommend it to anyone as an introduction to higher analysis. Perhaps the only objection that I might have to this book is the way that he segments his proofs. It seems to discretize what should be a well-flowing argument. Still, the proofs are excellent in spite of this barrier (though some might consider it an asset).

A Readable Introduction
This is an extremely readable introduction to the subject of calculus on arbitrary surfaces or manifolds. The author develops the subject from the beginning assuming only basic calculus and linear algebra - and then introduces concepts of integration and tensor analysis as the book progresses. Each segment is accompanied by a series of problems that does well to reinforce concepts. All in all, a good introduction. ... Read more

Book Description

Metric theory has undergone a dramatic phase transition in the last decades when its focus moved from the foundations of real analysis to Riemannian geometry and algebraic topology, to the theory of infinite groups and probability theory. The new wave began with seminal papers by Svarc and Milnor on growth of groups and the spectacular proof of the rigidity of lattices by Mostow. This was followed by the creation of the asymptotic metric theory of infinite groups by Gromov. The structural metric approach to the Riemannian category, tracing back to Cheeger's thesis, pivots around the notion of the Gromov-Hausdorff distance between Riemannian manifolds. This distance organizes Riemannian manifolds of all possible topological types into a single connected moduli space, where convergence allows the collapse of dimension with unexpectedly rich geometry, as revealed in the work of Cheeger, Fukaya, Gromov and Perelman. Also, Gromov found metric structure within homotopy theory and thus introduced new invariants controlling combinatorial complexity of maps and spaces, such as the simplicial volume, which is responsible for degrees of maps between manifolds. During the same period, Banach spaces and probability theory underwent a geometric metamorphosis, stimulated by the Levy-Milman concentration phenomenon, encompassing the law of large numbers for metric spaces with measures and dimensions going to infinity. The first stages of the new developments were presented in a course delivered by Gromov in Paris, and turned into the famous "Green Book" by Lafontaine and Pansu (1979). The present English translation of that work has been enriched and expanded with new material to reflect recent progress. Additionally, four appendices-by Gromov on Levy's inequality, by Pansu on "quasiconvex" domains, by Katz on systoles of Riemannian manifolds, and by Semmes overviewing analysis on metric spaces with measures-as well as an extensive bibliography and index round out this unique and beautiful book. ... Read more

Reviews (1)

The nature of high dimensions: a geometric insight
Formally speaking, this is the second edition of a set of Paris lecture notes published by Gromov two decades ago in the French language. However, such a wealth of entirely new material has been added that in essence weare talking of a new book.

Among the additions, the bulky new chapter 31/2+ stands out, dealing with the phenomenon of concentration of measure onhigh-dimensional structures. This is a relatively recent discovery ofmodern analysis and geometry, tracing its origin to the work of Paul Levyand especially Vitali Milman. The essence of the phenomenon is that on manymultidimensional structures, every `nice' function is constant with highprobability. The manifestations of the phenomenon are many - from geometricfunctional analysis (Dvoretzky theorem) through information theory(blowing-up lemma) and probability (law of large numbers) to graph theory(superconcentrators) and topological dynamics. As Gromov stresses in hisbook, even deeper aspects of the concentration phenomenon have been longsince discovered and are constantly explored in statistical physics in thecontext of phase transitions of various kind, and some of the first knownexamples where phase transitions appear in the context of geometry havebeen discovered by Gromov himself, e.g. for hyperbolic groups. Finding andexploring more instances of phase transitions in mathematics might wellbecome a unifying heuristic principle across a large number ofdisciplines.

The mathematical setting for dealing with concentration andrelated issues is the concept of a metric space equipped with finitemeasure, what Gromov calls an mm-space. Apart from concrete objects (suchas for instance spheres and cubes), there are `higher-level' examples ofmm-spaces, for instance those whose elements are isomorphism classes ofmathematical objects themselves (e.g. Riemanning manifolds or finitelygenerated groups). This leads to a probabilistic treatment of such objects.Of course Gromov's strength is that his treatment is always concrete and henever theorizes without having particular objects and applications inmind.

It is quite safe to claim that the full range and power ofapplications of the interaction between metric and measure are yet to bediscovered, which is what makes this book so important. It is rich in openquestions and suggested new research directions, but more than that, ithelps the reader to develop a good intuitive feeling of where things aregoing these days, what things ought to be done, and what constitutes propermathematics.

Even though I unexpectedly found myself among the privilegedones who received a copy of the book as a gift from the author, I wouldhave certainly purchased it otherwise, as I firmly believe that everymathematical library in the world, be it that of a top-class University orjust a modest, lovingly selected office collection of a humblemathematician, will be wanting without a copy of the monograph underreview, which might well become one of the most important books inmathematical sciences for the early XXIst century. ... Read more

Book Description

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

Reviews (5)

Yet another Popular (YAP) Math text
Warning: This book has no exercises

This book basically sucks.It has been hyped up for political reasons.It could make for a year long course in Differential Geometry if exercises were added to it, but as it stands, it's just a weak reference at best to introductory material.

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

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

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

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

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

Concise and To the Point!
I read the Chinese version of this book published by Peking University Press some time ago. Now, I am inpatiently waiting for the English version. It has been delayed again and again. This book is very concised and notsuitable for beginners who want to learn the subject by this book. Wu'sChinese book of "Rieman Geometry" is a more friendly book dealingon the introductory level. Nevertheless, I like Chern's book a lot. Itcomes handy to review the subject. If you are an advanced reader, you willfind this book to be very enjoyable like me. ... Read more

Book Description

The ideas of Elie Cartan are combined with the tools of Felix Klein and Sophus Lie to present in this book the only detailed treatment of the method of equivalence. An algorithmic description of this method, which finds invariants of geometric objects under infinite dimensional pseudo-groups, is presented for the first time.

As part of the algorithm, Gardner introduces several major new techniques.In particular, the use of Cartan's idea of principal components that appears in his theory of Repere Mobile, and the use of Lie algebras instead of Lie groups, effectively a linear procedure, provide a tremendous simplification. One must, however, know how to convert from one to the other, and the author provides the Rosetta stone to accomplish this. In complex problems, it is essential to be able to identify natural blocks in group actions and not just individual elements, and prior to this publication, there was no reference to block matrix techniques.

The Method of Equivalence and Its Applications details ten diverse applications including Lagrangian field theory, control theory, ordinary differential equations, and Riemannian and conformal geometry.

This volume contains a series of lectures, the purpose of which was to describe the equivalence algorithm and to show, in particular, how it is applied to several pedagogical examples and to a problem in control theory called state estimation of plants under feedback. The lectures, and hence the book, focus on problems in real geometry.

Special Features:

- only book to treat this subject in such depth and to include the algorithm, the use of principal components, and the use of infinitesimal analysis on the Lie algebra level.

Audience:

The reader should possess a background in calculus on manifolds and a familiarity with abstract linear and multilinear algebra.A first course on manifolds with an introduction to Lie groups and the ability to believe advanced results would be even more helpful. This volumes contains asides at various levels involving representation theory of Lie groups and Lie algebras, exterior differential systems, and Lie pseudo-groups. ... Read more

Book Description

The Baum-Connes conjecture is part of A. Connes' non-commutative geometry program. It can be viewed as a conjectural generalization of the Atiyah-Singer index theorem, to the equivariant setting (the ambient manifold is not compact, but some compactness is restored by means of a proper, co-compact action of a group G).

Like the Atiyah-Singer theorem, the Baum-Connes conjecture states that a purely topological object coincides with a purely analytical one. For a given group G, the topological object is the equivariant K-homology of the classifying space for proper actions of G, while the analytical object is the K-theory of the C*-algebra associated with G in its regular representation.

The Baum-Connes conjecture implies several other classical conjectures, ranging from differential topology to pure algebra.

This book is intended for graduate students and researchers in geometry (commutative or not), group theory, algebraic topology, harmonic analysis, and operator algebras. It presents, for the first time in book form, an introduction to the Baum-Connes conjecture. It starts by defining carefully the objects in both sides of the conjecture, then the assembly map that connects them. Thereafter, it illustrates the main tool to attack the conjecture (Kasparov's theory), and it concludes with a rough sketch of V. Lafforgue's proof of the conjecture for co-compact lattices in Sp(n,1), SL(3,3) and SL(3,"). ... Read more

Book Description

Hardbound. Computational Geometry is an area that provides solutions to geometric problems which arise in applications including Geographic Information Systems, Robotics and Computer Graphics. This Handbook provides an overview of key concepts and results in Computational Geometry. It may serve as a reference and study guide to the field. Not only the most advanced methods or solutions are described, but also many alternate ways of looking at problems and how to solve them. ... Read more

Book Description

The second edition featured a new chapter with a systematic development of variational problems from quantum field theory, in particular the Seiberg-Witten and Ginzburg-Landau functionals. This third edition gives a new presentation of Morse theory and Floer homology that emphasises the geometric aspects and integrates it into the context of Riemannian geometry and geometric analysis. It also gives a new presentation of the geometric aspects of harmonic maps: This uses geometric methods from the theory of geometric spaces of nonpositive curvature and, at the same time, sheds light on these, as an excellent example of the integration of deep geometric insights and powerful analytical tools. These new materials are based on a course at the University of Leipzig, entitled Geometry and Physics, attended by graduate students, postdocs and researchers from other areas of mathematics. Much of this material appears for the first time in a textbook. ... Read more

Reviews (1)

Intro to Riemannian Geom. and Geom. Analysis
Covers standard material on Reimannian Geometry. In addition: variational problems from QFT. Spin geometry and Dirac operators are explained in detail. ... Read more

Book Description

Recent years have seen the appearance of several books bridging the gap between mathematics and physics; most are aimed at the graduate level and above. Symmetry in Mechanics: A Gentle, Modern Introduction is aimed at anyone who has observed that symmetry yields simplification and wants to know why.

The monograph was written with two goals in mind: to chip away at the language barrier between physicists and mathematicians and to link the abstract constructions of symplectic mechanics to concrete, explicitly calculated examples. The context is the two-body problem, i.e., the derivation of Kepler's Laws of planetary motion from Newton's laws of gravitation.

After a straightforward and elementary presentation of this derivation in the language of vector calculus, subsequent chapters slowly and carefully introduce symplectic manifolds, Hamiltonian flows, Lie group actions, Lie algebras, momentum maps and symplectic reduction, with many examples, illustrations and exercises. The work ends with the derivation it started with, but in the more sophisticated language of symplectic and differential geometry.

For the student, mathematician or physicist, this gentle introduction to mechanics via symplectic reduction will be a rewarding experience. The freestanding chapter on differential geometry will be a useful supplement to any first course on manifolds. The book contains a number of exercises with solutions, and is an excellent resource for self-study or classroom use at the undergraduate level. Requires only competency in multivariable calculus, linear algebra and introductory physics. ... Read more

Reviews (2)

Poorly written
This is one of the very few books which I returned for refund.
The subject is intrinsicly interesting, and there is a need for a serious introductory text addressing the subject of geometry and physics. This one badly falls short, - carelessly written, with numerous irrelevant asides. She seems even to fail to realize that there exist three distinct geometric solutions to the Kepler problem. The bound, elliptic case is only one.
This book has supposedly been written for high-undergraduate students or early-year graduate students. It serves neither adequately.

A welcome book
There are a number of books available on the "geometric" view of physics (Classical Mathematical Physics, by Thirring, The Geometry of Physics, by Frankel, and Foundations of Mechanics, by Abraham & Marsden). The size, level of sophistication and extensive background assumed by these books can be very intimidating. On the other hand, the subject "looks" beautiful, and the benefits of using geometric intuition are desirable to many people.

Singer's book stands class of its own in these respects. All the basics of the geometrical "machinery" are there, in a book that is only 224 pages in length. Chapter one starts with a standard derivation of the equations of the "two-body planetary motion" problem; subsequent chapters proceed to introduce the necessary modern geometrical and mathematical concepts (differential geometry). The final chapter then revisits the "planetary motion" problem using the modern concepts previously introduced. Excellent!

There are some misprints, but the author has a Web page of errata. The book has numerous exercises, with many solutions included. I find myself rereading parts of this book over and over.

Reader be warned; the concepts are new, and it does take work to internalize them. However, this is the most accessible book on the subject available, and also one of the most affordable. The author references many other books, for the reader who wants to go further in the mastery of this subject (one excellent book which is not mentioned, however, is "Differential Forms: A Complement to Vector Calculus", by Weintraub). Enjoy! ... Read more

Book Description

This book is an exposition of what is currently known about thefundamental groups of compact Kähler manifolds.

This class of groups contains all finite groups and is strictlysmaller than the class of all finitely presentable groups. Forthe first time ever, this book collects together all the resultsobtained in the last few years which aim to characterise thoseinfinite groups which can arise as fundamental groups of compactKähler manifolds. Most of these results are negative ones,saying which groups do not arise. They are proved using Hodgetheory and its combinations with rational homotopy theory, with$L^2$ --cohomology, with the theory of harmonic maps, and withgauge theory. There are a number of positive results as well,exhibiting interesting groups as fundamental groups ofKähler manifolds, in fact, of smooth complex projectivevarieties.

The methods and techniques used form an attractive mix oftopology, differential and algebraic geometry, and complexanalysis. The book would be useful to researchers and graduatestudents interested in any of these areas, and it could be usedas a textbook for an advanced graduate course. One of itsoutstanding features is a large number of concrete examples.

The book contains a number of new results and examples which havenot appeared elsewhere, as well as discussions of some importantopen questions in the field. ... Read more

Book Description

This self-contained book by a leading topologist is devoted to algorithmic low-dimensional topology, a branch of mathematics that has recently been undergoing an intense development. The book contains plenty of important fundamental material, which is carefully presented. The book also contains some of the author's own original contributions. For the first time ever, it gives a full exposition of the complexity theory of 3-manifolds and a complete proof of the solution of the homeomorphism problem for Haken manifolds. The style of the book is very lively, with a lot of useful pictures, making the book enjoyable for those who like visual topology. The writing is clear and the proofs are careful and detailed. This book fills a gap in the exisiting literature and will become a standard reference for this aspect of 3-dimensional topology both for graduate students and researchers. ... Read more

Book Description

This book is an expanded version of lectures given at a summer school on symplectic geometry in Nordfjordeid, Norway, in June 2001. The unifying feature of the book is an emphasis on Calabi-Yau manifolds. The first part discusses holonomy groups and calibrated submanifolds, focusing on special Lagrangian submanifolds and the SYZ conjecture. The second studies Calabi-Yau manifolds and mirror symmetry, using algebraic geometry. The final part describes compact hyperkahler manifolds, which have a geometric structure very closely related to Calabi-Yau manifolds. The book is an introduction to a very active field of research, on the boundary between mathematics and physics. It is aimed at graduate students and researchers in geometry and string theory and intended as an introductory text, requiring only limited background knowledge. Proofs or sketches are given for many important results. Moreover, exercises are provided. ... Read more

Book Description

This is the new edition of Serge Lang's "Differential and Riemannian Manifolds." This text provides an introduction to basic concepts in differential topology, differential geometry, and differential equations, and some of the main basic theorems in all three areas: for instance, the existence, uniqueness, and smoothness theorems for differential equations and the flow of a vector field; the basic theory of vector bundles including the existence of tubular neighborhoods for a submanifold; the calculus of differential forms; basic notions of symplectic manifolds, including the canonical 2-form; sprays and covariant derivatives for Riemannian and pseudo-Riemannian manifolds; applications to the exponential map, including the Cartan-Hadamard theorem and the first basic theorem of calculus of variations. ... Read more

Reviews (1)

Complete but lacking unity
This book looks more like a big collection of essays than really a treatise on one subject. If you have read other books by Lang you will notice that some material included in previous works appears reproduced here word by word. Also, there are some inconsistencies in the notation of different sections, making it obvious that different parts of the book were written at different times, and perhaps by different persons.

All these shortcomings don't mean that the book is bad. Quite the opposite: It is a very complete survey on modern differential geometry, including from the fundamentals up to recent results. The graduate student and the working mathematician will find it very useful. ... Read more

Book Description

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

Book Description

The theory of minimal surfaces has expanded in many directions over the past decade or two. This volume gathers in one place an overview of some of the most exciting developments, presented by five of the leading contributors to those developments. H.Fujimoto, who obtained the definitive results on the Gauss map of minimal surfaces, reports on Minimal Surfaces and Nevanlinna Theory. S.Hildebrandt provides an up-to-date account of the Plateau problem and related boundary-value problems. D.Hoffman and H.Karcher describe the wealth of results on embedded minimal surfaces from the past decade, starting with Costa's surface and the subsequent Hoffman-Meeks examples. Finally, L.Simon covers the PDE aspect of minimal surfaces, with a survey of known results both in the classical case of surfaces and the higher dimensional case. ... Read more