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How are the great pyramids like seashells? Ask mathematician Midhat J. Gazale, then brace yourself for a heady ride through the wilds of self-similar geometry in Gnomon: From Pharaohs to Fractals, his paean to the roiling mysteries that lie beneath the tranquil surfaces of such objects. The great mathematician Hero of Alexandria defined a gnomon as an object that, when added to another, creates a new object similar in form to the original. Gazale, also of Alexandria, goes much further and uses 20th-century concepts to fully explore "gnomonicity"--the property of self-similarity.

Be prepared for slow going: Gnomon is densely packed with information and concepts foreign to all but the professional mathematician, but Gazale's enthusiasm and brilliant illustrations win the day. Whether he's moving on from the familiar golden rectangle to his own "silver pentagon" or rooting around in the numbers underlying the groovy fractal images popping up on T-shirts worldwide, he takes care to explain to the reader not just what's going on mathematically but what all this abstraction really means to us. Few science books, and even fewer mathematics books, achieve that kind of depth. --Rob Lightner ... Read more

Reviews (4)

Well written, but tediously boring book
....I found this book to be relatively well written, but tediously boring. This book should probably be considered "Recreational Mathematics" and is jammed with math; very little prose accompanies it. Indeed, if you do not love math, or do not love math for the sake of math (rather than, say, a physicist who merely loves math for the sake of his primary interest in physics) then you will probably not enjoy this book.

The prose that the book contains seems to attempt to relate the various math that Gazale demonstrates to practical problems, especially in electrical engineering. I found these analogies to be difficult to follow and did not create much understanding of the material. For example, he discusses resonant LCR circuits and power transmission lines, and his descriptions pale in comparison to those I found in Feynman's "Lectures on Physics."

Now,... this book has a number of useful things in it which is why I say it's "well written." It does include mathematical summaries in the end of every chapter. In my opinion, this was the best part of the book. By reading only the summaries, the reader "takes away" most of what the author is trying to say, without having to suffer through the rest of it. If the summary is not self-evident, the reader can always go back to the chapter to divine how one of the entries in the summary was derived.

All in all, I would not recommend this book to those interested in anything other than math for math's sake. If your interest lies elsewhere, or even only obliquely in math (such as in learning math for a specific use), then avoid this book. However, if you do enjoy a mathematical romp, then this book could well be the one for you.

Slow going, but worth it
According to Gazale', "Hero of Alexandria defined the gnomon as that figure (a number or a geometric figure) which, when added to another figure, results in a figure similar to the original." Gazale's book is, therefore, about self-similarity in numbers and geometry.

The subject sounds simple enough, but I found this to be a pretty tough book. That might be partly due to the fact that I've always had a hard time focusing my attention on number theory. This book has a lot of basic stuff about numbers, and I found much of that subject rather tedious and (dare I say it?) boring. I know that's an ignorant thing to say - after all, mathematics is a beautiful subject in its own right, and there is some really neat stuff in number theory. But it was still a tough book for me to wade through.

The introduction is mostly historical background, and a little truncated. It serves primarily to illustrate a few basic concepts in self-similarity. The author continues this theme with a short description of figurate and m-adic numbers. Gazale tends to use more technical language than many casual readers are likely to recognize. Yet this really isn't a book on formal mathematics, either. It's really somewhere in between.

Gazale often draws on themes from Martin Gardner's series of articles in Scientific American, and in some ways, his book reflects Gardner's style. And, while much of this book seems focused on abstract details, there are occasional forays that illustrate amazing connections between what looks like pure mathematics and the real world.

Chapter 2, titled "Continued Fractions," is foundational. I really enjoyed this section, and think the book is worth having for this chapter alone. Beginning with Euclid's algorithm, Gazale offers a natural introduction to continued fractions. Then, in his characteristic style, he continues to explore every nook and cranny of this fascinating branch of mathematics. Among the most pleasing results of this chapter is his demonstration of the mirrored similarity in the appearance of numbers as they are represented by continued fractions, and as they are represented by our traditional positional number system. For example, he shows that both representations are always convergent and uniquely correspond to a number. However, while infinite periodic representations correspond to rational numbers in the positional system, they correspond to quadratic irrationals in the system of continue fractions. And, while transcendental and irrationals are infinite nonperiodic representations in both systems, there are some beautiful expressions of some transcendental numbers in the system of continued fractions that left me mesmerized.

One particularly nice feature is the way the author summarizes the important equations at the back of each chapter. Some of these summaries are several pages long, and they actually do a good job of encapsulating the essential material. In fact, the summaries are so well done that, if you read the book, you probably will be able to go back and use the formulas in the back of the chapters without having to refer back to the text.

If you ever wanted to know about the Fibonacci sequence, I can hardly imagine a book that will satisfy you better than this one. The first thing you will learn is that the Fibonacci sequence you met in grade school is just a small subset of a more general form. Then, in a whirlwind of mathematical activity you will see the general recursive formula (which depends, of course, on the seed and gnomonic numbers). This is followed by explicit formulae for the terms of the sequence and even a demonstration of how some of these equations, in the limit, model the behavior of wave propagation in an electronic transducer ladder, and the movement of a ganged series of pulleys.

A continuing source of amazement is the way in which the mathematical themes in this book are so interconnected. That's fitting, I suppose, for a book called Gnomon. In the chapter on whorled figures we see many of the other subjects in this book reappearing.

The book also has an excellent chapter on the golden number, and another on the silver number. The golden number, as you may know, shows up all over the place, and not just in Gazale's book. Here, he connects it with the Fibonacci sequence, whorled spirals, and golden rectangles. And for every example using the golden number, not surprisingly, there is another using the silver number. It's fascinating to read of all the ways in which these numbers shows up, and try to contemplate the underlying order that makes it happen this way.

Things get a little more abstract toward the end of the book (but no less interesting), and a bit harder to read. There are some very interesting developments with spirals and the rotation matrix, along with some interesting construction techniques for making your own spirals with paper and pen.

The last chapter, on fractals, exercised my little gray cells more than the rest of the book. This is not your typical discussion about fractals, with pretty pictures and non-technical explanations about self-similarity at any level. While Gazale does not dive in with the sort of mathematical rigor to which a pure mathematician would aspire, he claims to have written an unusual chapter on the subject, derived directly from number-theoretic considerations. This chapter will keep you busy following all the ins and outs of some pretty involve matrix mathematics. If you like the fast-Fourier algorithm, I think you'll love it.

This book is definitely not for everyone. But if you really, really like mathematics, and especially number-theoretic mathematics, I think you will like it. It will most definitely exercise your mind, but then again, that's what a good book on mathematics is supposed to do. Isn't it?

It would be better if this guy could write
I'm a regular reader of all sorts of books on math, and so Gnomon seemed a natural for me. I have a master's degree in computer science (bachelors, too, but i digress) and this sort of thing is right up my alley. The book doesn't really cover any new ground, but it does gather separate things into one volume, which makes it nice as a reference. The biggest problem with the book is with the actual text. This guy can't write. Yes, the material is technical, yes it's slow going, but that is no excuse for poorly structured arguments and incoherent organization of the material.

It's all here, but you'll need to work through it slowly and try to infer what he means because he leaves out a bunch of foundation work.

An extremely original book , full of ideas and discoveries.
A very approachable text that appeals to the academic as well as non academic.The simplicity and power of mathematics is demonstrated by this erudite author who promotes this unique and historical approach of the evolution of math. He successfully descibes the self similar processes in math as well as in life forms. Self similarity is the common thread. Very stimulating. ... Read more

Book Description

This is an introduction to noncommutative geometry, with special emphasis on those cases where the structure algebra, which defines the geometry, is an algebra of matrices over the complex numbers. Applications to elementary particle physics are also discussed. This second edition is thoroughly revised and includes new material on reality conditions and linear connections plus examples from Jordanian deformations and quantum Euclidean spaces. Only some familiarity with ordinary differential geometry and the theory of fiber bundles is assumed, making this book accessible to graduate students and newcomers to this field. ... Read more

Reviews (2)

Very nice, lots of good stuff
This book is (partially) the answer to my prays: an introductory book on noncommutative geometry, something I've been waiting since I discovered the topic in Connes' seminal text, which I've also reviewed here. Instead of exposing the historical origins, then firing a goddamn chaingun of advanced topics (something quite fascinating, because of the potential of the theory, but not pedagogical), Madore uses a more friendly way of exposing things, by mantaining a compromise between the most natural motivations to the techniques of the subject and the places where the background needed is not so overwhelming. He do teach much of the background (in the sense that you don't need to master functional analysis, operator algebras and advanced differential geometry), but he goes quite fast on it, requiring a rather mature mathematical mind. As noncommutative geometry is not for the faint of the heart, I guess he's not asking too much after all.

The pedagogy of the book is also benefitted from the post-"Connes' book" evolution of noncommutative geometry, because in 1999 the theory and its (real and potential) applications were a great deal more mature and solid than in 1994. Being this theory a work in progress, the better the math knowledge the reader has, the more he or she will learn from Madore's book, which stands maybe as the only pedagogical exposition of noncommutative geometry (now I'm waiting for the huge book from Garcia-Bondia and his colaborators, to be published by Birkhauser in 2001, hope that it contains more background; it would be very useful for those interested in beginning research on the area).

An Introduction to Noncommutative Differential Geometry and
FOR PHYSICIST, I strongly reccomend this book! There are so many physical examples in this book. Always we physicists hate mathamatical proofs like a torture. But this book concentrates applications to physics. If you want to study Noncommutative Geometry as a physicist, this book should be chosen as the first introduction! ... Read more

Book Description

This book contains essential material that every graduate student must know. Written with Serge Lang's inimitable wit and clarity, the volume introduces the reader to manifolds, differential forms, Darboux's theorem, Frobenius, and all the central features of the foundations of differential geometry. Lang lays the basis for further study in geometric analysis, and provides a solid resource in the techniques of differential topology. The book will have a key position on my shelf.

-Steven Krantz, Washington University in St. Louis

This is an elementary, finite dimensional version of the author's classic monograph, Introduction to Differentiable Manifolds (1962), which served as the standard reference for infinite dimensional manifolds. It provides a firm foundation for a beginner's entry into geometry, topology, and global analysis. The exposition is unencumbered by unnecessary formalism, notational or otherwise, which is a pitfall few writers of introductory texts of the subject manage to avoid. The author's hallmark characteristics of directness, conciseness, and structural clarity are everywhere in evidence. A nice touch is the inclusion of more advanced topics at the end of the book, including the computation of the top cohomology group of a manifolds, a generalized divergence theorem of Gauss, and an elementary residue theorem of several complex variables. If getting to the main point of an argument or having the key ideas of a subject laid bare is important to you, then you would find the reading of this book a satisfying experience. ... Read more

Book Description

The geometrical view of mechanics is based on the study ofcertain exterior systems, the most classical of which are Pfaffiansystems. In this book, we present the classification theorems(Frobenius, Darboux) and the local classification of Pfaffian systemsof five variables, following Cartan. We also present a new class of exterior systems, calledk-symplectic systems, generalizing the notion of symplecticform. These systems permit us to write in the language of exteriorforms the equations proposed by Nambu for a model of statisticalmechanics. Audience: This book is aimed at graduate students and atresearch workers in the field of mathematics, differential geometry,statistical mechanics, mathematics of physics and Lie algebras. ... Read more

Reviews (1)

An extremely weak book
This book intends to give new insights of Nambu mechanics, but up to some calculations in differential forms (in sense of Cartan, but were prolongations and other fundamental topics are not treated), authors do not provide any sustainable physical result. It seems obvious that they are far from having expertise on physics. There is no mention to multibrackets in this book, which is the well known technique to attack the problem. ... Read more

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Reviews (2)

Accessibility, Understandabilityto Balance Theorem Proving
In my experience accessibility and understandability have to balance theorem proving.
This book is well on it's way to becoming a modern classic even if it is somewhat dated
in it's material. Robert Osserman is recognized as a great teacher of mathematics
by many in the mathematics community. But his approach has come between
the seeker of knowledge and his goal of understanding and mastery of the subject.
It is not his fault as in a peer review he would be generally
rewarded for his approach as he has been in his paper
publishing.But I can see grad students rushing to resell this book
as it has a limited amount of useful reference information
and very few of the actual definitions of the curves talked about.
The references are classic and many of the people are dead or retired by now.
It is access to the actual formulations for the minimal surfaces and explanations that
help in understanding the material not by theorem proving ,
but by visualization and explanation that are needed.
In that: this book fails. But compared to many other such mathematics books , it is actually superior!

A classical source
Osserman's survey is both an excellent starting point for beginners and a reference for some interesting problems in the subject (although it was written some years ago!). Weiesrtrass' classical representation of minimal surfaces, their topology...how can all these be in such a small book? ... Read more

Reviews (1)

An extraordinary book!
This book describes how to tabulate knots with the help of computers. You could ask, so, what is special about that and why is this difficult? It is difficult because a knot or link is a topological object and hence, it is not clear how to represent such an object that it can be processed by an computer at all. Even if you know the Dowker-Thistlethwaite notation it is still an art to do it.
In my opinion this book is ahead of its time, because the field of computational topology/knot theory is still in its infacncy. ... Read more

Book Description

This introduction to the conformal differential geometry of surfaces and submanifolds covers those aspects of the geometry of surfaces that only refer to an angle measurement, but not to a length measurement. Different methods (models) are presented for analysis and computation. Various applications to areas of current research are discussed, including discrete net theory and certain relations between differential geometry and integrable systems theory. ... Read more

Book Description

The uniqueness of this text in combining geometric topology and differential geometry lies in its unifying thread: the notion of a surface. With numerous illustrations, exercises and examples, the student comes to understand the relationship between modern axiomatic approach and geometric intuition. The text is kept at a concrete level, 'motivational' in nature, avoiding abstractions. A number of intuitively appealing definitions and theorems concerning surfaces in the topological, polyhedral, and smooth cases are presented from the geometric view, and point set topology is restricted to subsets of Euclidean spaces. The treatment of differential geometry is classical, dealing with surfaces in R3 . The material here is accessible to math majors at the junior/senior level. ... Read more

Reviews (4)

a remark on omissions
I have not read the book, only the reviews.In one excellent review here it is remarked that it is "unfortunate" that the author does not prove the Schoenflies theorem and the triangulability of surfaces.

later this same reviewer observes that the proof of the smooth Gauss Bonnet theorem in the book seems relatively hard.I merely wish to point out that the author has made choices in the reader's interest both by what he includes and what he omits.

The two theorems named above which are not proved, could well take another entire book to prove.They are far harder than the smooth Gauss Bonnet theorem.

I have seen entire books devoted to proving triangulability, and Schoenflies theorem was the subject of weeks of tedious work in a topology course I took as a student.I still dislike even hearing of this result.So if these omissions are the reviewer's only criticisms of the book, they should rightly be considered pluses.

Hence I also give the book at least 4 stars, by logical deduction.

Good introduction
This book is suitable for reading at an advanced undergraduate or beginning graduate level. The author is careful to present the subject from both a rigorous point of view and one that emphasizes the geometric intuition behind the subject. These two approaches to teaching topology are not mutually exclusive, with this book giving a good example of this.

After a brief overview of the elementary topology of subsets of Euclidean space in chapter 1, topological surfaces are discussed in chapter 2. Surfaces are built up from arcs, disks, and one-spheres. Unfortunately, the proofs of the theorem of invariance of domain and the Schonflies Theorem are not included, but references are given. Gluing techniques though are effectively discussed, and the author does not hesitate to use diagrams to explain the relevant concepts. The more popular constructions in surface topology, namely the Mobius strip and the Klein bottle are given as examples of the cutting and pasting techniques. The amusing fact that the Klein bottle can be obtained from gluing two Mobius strips along their boundaries is proven.

The theory of simplicial surfaces is discussed in the next chapter. Simplicial surfaces are much easier to deal with for beginning students of topology. Simplicial complexes are introduced first, and the author then studies which simplicial complexes have underlying spaces that are topological surfaces. He proves that this is the case when each one-dimensional simplex of the complex is the face of precisely two two-dimensional simplices, and the underlying space of each link of each zero-dimensional simplex of the complex is a one-dimensional sphere. Unfortunately, the author does not prove that any compact topological surface in n-dimensional Euclidean space can be triangulated. The Euler characteristic is defined first for 2-complexes and it is shown that it is the same for two simplicial surfaces that triangulate a compact topological surface. The author does prove in detail the classification of compact connected surfaces. Interestingly, the author also proves a simplicial analogue of the Gauss-Bonnet theorem, and gives a proof of the Brouwer fixed point theorem.

The author turns to smooth surfaces in the next few chapters, wherein curves are defined along with the relevant differential-geometric notions such as curvature and torsion. The fundamental theorem of curves is proven. The reader is first introduced to the concept of what in more advanced treatments is called a differentiable manifold, and several concrete examples are given of smooth surfaces. The differential geometry of smooth surfaces is outlined, with the first fundamental form and directional derivatives discussed in great detail. The reader should be familiar with the inverse function theorem to appreciate the discussion of regular values.

Even more interesting differential geometry is discussed in chapter 6, which covers the curvature of smooth surfaces. The important Gauss map is defined, along with the Weingarten map and the second fundamental form. This allows an intrinsic notion of curvature, but the author does perform explicit computations of curvature using various choices of coordinates. The proof that Gaussian curvature is intrinsic (Theorema Egregium) is proven, along with the fundamental theorem of surfaces. Geodesics, so important in physical applications, are discussed in the next chapter. The reader gets a first look at the "Christoffel symbols", even though they are not designated as such in the book.

The book ends with a thorough treatment of the Gauss-Bonnet theorem for smooth surfaces. The smooth case is much more difficult to prove than the simplicial case, as the reader will find out when studying this chapter. The author also gives a very brief introduction to non-Euclidean geometry.

Presentation of The Spirit...
Lots of times the mathematicians stuck in proofs,in that fool symbols, forgetting the ideas, the picture.One can never find the right way withclosed eyes.This book teaches to think, getting beyond the symbols. Ithas also useful advises about the research areas. The author made his Phdat Cornell with D.Henderson.A beautiful undergraduate text.

Just a mediocre book of lesser extent
First, the title reads fine. But there's a catch. This kind of title sounds like it covers all. The truth is. It ain't true. Second. The author's attitude. I'd rather say the author is talking to himself. ... Read more

Book Description

This book is an introduction to the topological rigidity theorem for compact non-positively curved Riemannian manifolds. It contains a quick informal account of the background material from surgery theory and controlled topology prerequesite to this result. It is intended for researchers and advanced graduate students in both differential geometry and topology. This book is the content of a course given by the author at TIFR in 1993. ... Read more

Book Description

This book is primarily a textbook introduction to various areas of discrete geometry. In each area, it explains several key results and methods, in an accessible and concrete manner. It also contains more advanced material in separate sections and thus it can serve as a collection of surveys in several narrower subfields. The main topics include: basics on convex sets, convex polytopes, and hyperplane arrangements; combinatorial complexity of geometric configurations; intersection patterns and transversals of convex sets; geometric Ramsey-type results; polyhedral combinatorics and high-dimensional convexity; and lastly, embeddings of finite metric spaces into normed spaces. ... Read more

Book Description

The only book that introduces differentialgeometry through a combination of an intuitive geometricfoundation, a rigorous connection with the standard formalisms,computer exercises with Maple, and a problems-based approach. Starting with basic geometric ideas, DifferentialGeometry uses basic intuitive geometry as a starting point tomake the material more accessible and the formalism moremeaningful. The book presents topics through problems to providereaders with a deeper understanding. And, it introduces hyperbolicgeometry in the first chapter rather than in a closing chapter as inother books.An important reference and resourcebook for any reader who needs to understand the foundations ofdifferential geometry. ... Read more

Reviews (1)

Fantastic concept, flawed execution
It's certainly a great concept: explain differential geometry - and andthe myriad real-world applications of this subject - by appealing to ourgeometric intuition! (Those who have read just about any other text willrealize that I am sincere, not sarcastic in this remark - the intuitiveapproach is quite unusual in treatments of the subject material.)

To alimited degree, the book is a success. The first chapter flows rathersmoothly, and could actually be used to introduce differential geometry inan advanced high school classroom. I would consider that in and over itselfto be a truimph! In places, it's fun to read, and some of the"constructions" (often using three dimensions) are both cleverand helpful. And I must confess that reading this book I picked up bits andpieces of intuition that I had missed when reading other texts.

For allof these reasons, I found myself really wanting to like this book; sadly, Iultimately found that I could not. Unfortunately, the intuitive approachstarts to break down as the book proceeds.In the later chapters, I couldonly intuitively grasp and fully understand what Henderson was trying toexplain because of previous familiarity with the material; I would havepretty baffled without prior knowledge of the subject. The writing andpresentation just does not compare with that in some of the better (if moretraditional) texts in differential geometry, such as Manfredo P. Do Carmo'sDiffertial Geometry of Curves and Surfaces or Michael Spivak's excellentfive-volume Comprehensive Introduction to Differential Geometry. If one isfamiliar with those (or other similar) texts, it might be fun to take alook at Henderson's book. If not, look there first - or at least look thereas well - in your explorations of this field of mathematics. ... Read more

Book Description

Non-compact Riemannian symmetric spaces and their quotients byarithmetically defined groups of isometries occur in many parts ofmathematics. For many purposes it has been necessary to compactifythem. This monograph attempts to give a systematic, and in part newexposition of these compactifications, of new ones, theirinterrelations and of the context out of which they arose.Compactifications of the more general semisimple symmetric spaces arealso considered. The book is divided into three main parts. Part I is devoted tofive types of compactifications, some related even isomorphic, all G-spaces, of the quotient X=3DG/K of a semisimple linear real Lie group Gwith finitely many connected components by a maximal compact subgroupK. The second part treats compactifications of the quotientsGamma\X, where Gamma is an arithmetic subgroup of G, assumed to bedefined over the field Q of the rational numbers. In the third part,three new types of compactifications are examined: the directconstructions of T. Oshima and T. Oshima--T. Sekiguchi, the gluing ofa certain number of copies of a compact manifold with corners, and thereal points of the so-called wonderful compactification of thecomplexification of X, or more generally G/H. The compactification of noncompact Riemannian symmetric spacesleads to a rich area of research in which many mathematicaldisciplines come together: algebraic topology, geometry, number theoryand representation theory. Familiarity with the theory of realsemisimple Lie groups and symmetric spaces, and in Part II, of linearalgebraic groups over Q is assumed although much material is recalledalong the way. Of interest and use to researchers and graduatestudents in Lie Theory or Representation Theory. ... Read more

Book Description

Starting at an introductory level, the book leads rapidly toimportant and often new results in synthetic differential geometry.From rudimentary analysis the book moves to such important results as:a new proof of De Rham's theorem; the synthetic view of global action,going as far as the Weil characteristic homomorphism; the systematicaccount of structured Lie objects, such as Riemannian, symplectic, orPoisson Lie objects; the view of global Lie algebras as Lie algebrasof a Lie group in the synthetic sense; and lastly the syntheticconstruction of symplectic structure on the cotangent bundle ingeneral. Thus while the book is limited to a naive point of viewdeveloping synthetic differential geometry as a theory in itself, theauthor nevertheless treats somewhat advanced topics, which are classicin classical differential geometry but new in the synthetic context.Audience: The book is suitable as an introduction to syntheticdifferential geometry for students as well as more qualifiedmathematicians. ... Read more

Reviews (1)

A Useful and Easy Introduction
This book is by far the most readable introduction to Synthetic Differential Geometry that there currently is. The book concentrates on building up axiomatic SDG with hardly a reference to ways of modelling it (ie: using topos theory), leaving the complicated mathematics until a good feeling for how SDG behaves is acquired. A good first read. ... Read more

Reviews (1)

A superb job...packed full of insights
This book could be loosely characterized as an attempt to generalize the theory of Riemann surfaces to that of Riemannian manifolds. The reader familiar with the theory of Riemann surfaces will perhaps find this book easier to read than one who has not. But the author has not assumed that the reader has had any prior exposure to Riemann surfaces, and so the reader without such background will find the reading straightforward. The paradigm in the book is the connection between the topology of Riemannian manifolds and their metric geometry. It is the metric structure of Riemannian manifolds that is responsible for their fame, due especially to their use in physics. Through the use of de Rham cohomology, Hodge theory, and other techniques from differential geometry, the author shows how to give an overview of the intrinsic ("coordinate-free") global differential geometry of Riemannian manifolds and how that geometry is connected to its topology.

Chapter 1 is a review of elementary differential geometry that is to be used in the rest of the book. Then in chapter 2 the author begins with a review of singular homology and de Rham cohomology. The key point, proved in an appendix, is the de Rham theorem which establishes an isomorphism between de Rham and singular cohomology. The pth Betti number is then the number of linearly independent closed differential forms of degree p modulo the exact forms of degree p. The rest of the chapter is devoted to showing how this result was extended by the mathematician W.V.D Hodge to a restricted class of forms, the famous "harmonic forms". Now called Hodge theory, it is a homology theory based on the Laplace-Beltrami operator, which generalizes, as expected, Laplace's equation.

Chapter 3 is devoted to finding an explicit expression for the Laplace-Beltrami operator in local coordinates. This expression is dependent on the Riemannian curvature of the Riemannian manifold, and so the homology of a compact and orientable manifold will depend on its curvature. The issue then is finding harmonic forms of a given degree. The obstruction to the existence of these is given by a particular quadratic form involving the curvature tensor. The absence of harmonic forms of degree p gives that the pth Betti number is zero. In particular the author shows that the Betti numbers of a compact, orientable, conformally flat Riemannian manifold of positive definite Ricci curvature are all zero. The author then applies these results to compact Lie groups in chapter 4. The harmonic forms on compact Lie groups are those differential forms that are invariant under both left and right translations of the group. The author shows that the first and second Betti numbers of compact Lie groups are zero and shows the existence of a harmonic 3-form, the latter proving that the third Betti number is greater than or equal to one.

The author turns his attention to complex manifolds in chapter 5. He approaches these objects from the standpoint of first defining complex structures on separable Hausdorff spaces. The complex structures then allow a definition of a Riemannian metric on these spaces. If the metric does have any torsion, then one can associate a particular 2-form with the metric and the complex structure that is closed. This 2-form is the famous "Kaehler metric", and the resulting space is called a "Kaehler manifold". The local geometry of Kaehler manifolds is referred to as "Hermitian geometry", and the author studies in detail this geometry in this chapter. Loosely speaking, a Kaehler metric can be viewed as a generalization of "flatness" in the usual Riemannian case, for the author shows that at each point of a Kaehler manifold there exists a system of local complex coordinates which is geodesic. He also introduces the important concept of a holomorphic p-form, and shows that on a Kaehler manifold these are harmonic.

In chapter 6, the author studies in detail how curvature and homology are related for the case of Kaehler manifolds. The results in this chapter could be viewed as a generalization of the classical results concerning compact Riemann surfaces, namely that the universal covering space of a complex n-dimensional compact Kaehler manifold of constant holomorphic curvature K is a projective space for K > 0, the interior of a unit sphere for k < 0, and the space of complex variables for K = 0. After defining the holomorphic curvature, the author shows that the pth Betti number of a compact Kahler manifold M with positive constant holomorphic curvature is zero if p is odd and 1 if p is even. In addition, he shows that any holomorphic form of degree p, for p > 0 and p less than or equal to n, on a compact Kaehler manifold with positive definite Ricci curvature is zero. The author also gives the reader a taste of sheaf theory, in which he discusses briefly the Kodaira vanishing theorems.

In the last chapter, the author generalizes what was done in chapter 3 regarding conformal transformations on Riemannian manifolds, namely that an infinitesimal holomorphic transformation of a compact Kaehler manifold can be viewed as the solution of a system of differential equations which involve the Ricci curvature. Conditions are given for making this transformation an isometry, and the author shows that for a compact Kaehler manifold of complex dimension greater than 1, an infinitesimal conformal transformation is holomorphic if and only if it is an infinitesimal isometry. This leads him to consider the groups of holomorphic transformations, and he gives conditions under which a compact complex manifold cannot admit a transitive Lie group of holomorphic transformations. The author also studies the most general class of Riemannian manifolds for which an infinitesimal conformal transformation is also an infinitesimal isometry. These are the famous "almost Kaehler" manifolds, and the author shows that an infinitesimal conformal transformation of a compact almost Kaehler manifold of dimension 2n for n > 1 is an infinitesimal isometry. ... Read more

Book Description

This volume is the first in a projected series devoted to the mathematical and philosophical works of the late Claude Chevalley. It covers the main contributions by the author to the theory of spinors. Since its appearance in 1954, "The Algebraic Theory of Spinors" has been a much sought after reference. It presents the whole story of one subject in a concise and especially clear manner. The reprint of the book is supplemented by a series of lectures on Clifford Algebras given by the author in Japan at about the same time. Also included is a postface by J.-P. Bourguignon describing the many uses of spinors in differential geometry developed by mathematical physicists from the 1970s to the present day. An insightful review of "Spinors" by J. Dieudonne is also made available to the reader in this new edition. ... Read more

Book Description