## DIFFERENTIABLE MANIFOLDS BY YOZO MATSUSHIMA -A LOST CLASSIC ON SMOOTH MANIFOLDS FOR FIRST YEAR GRADUATE STUDENTS, NOW AVAILABLE!

YET ANOTHER CLASSIC OUT OF PRINT TEXTBOOK, NOW AVAILABLE AGAIN AS BOTH A KINDLE AND INEXPENSIVE PAPERBACK BY BLUE COLLAR SCHOLAR! THEREFORE, THIS WONDERFUL BOOK BECOMES ONCE MORE A STANDARD TEXT FOR GRADUATE STUDENTS!

BEST OF ALL, BOTH A BRAND NEW PREFACE & COMPREHENSIVE SUPPLEMENTARY BIBILIOGRAPHY HAS BEEN WRITTEN BY KARO MAESTRO AKA THE MATHEMAGICIAN!

### Reviews For Matsushima’s *Differentiable Manifolds*:

This book, originally published in Japanese (Shokabo, Tokyo 1965), provides an introduction to the theory of differentiable manifolds, Lie groups and homogeneous spaces. It is designed as an advanced undergraduate or an introductory graduate course. The book consists of five chapters, one of them being introductory. Chapter II introduces differentiable and complex manifolds with their fundamental structures: differentiable (holomorphic) functions, tangent spaces, vector fields etc. The main results proved here are Bard’s theorem, the definition of one-parameter transformation groups and the existence of partitions of unity on paracompact manifolds. Chapter III considers tensor fields and differential forms with the usual algorithms (exterior differentiation, Lie differentiation) and the classical results such as Frobenius’ theorem, Foincare’s lemma etc…….

..The integration of differential forms is exposed in chapter V, where one studies orientation problems ‘Stokes’ theorem and the degree of mappings. The fourth chapter of the book is …….an introduction to Lie groups and homogeneous spaces and contains the fundamental results in this field (topological groups, Lie groups and Lie algebras, subgroups, quotient spaces etc.). The book is carefully written with detailed proofs (including some which one rarely finds in textbooks), examples and problems.

-I. Vaisman, Zentralblatt MATH

### More Reviews For Matsushima’s *Differentiable Manifolds*:

Careful, conventional introduction to differentiable manifolds, Lie groups and differential forms. Presumes and reviews elementary topology, linear algebra and multivariable calculus.…..it is a valuable exposition, accessible to anyone beginning study in this field.

-L.A.S.,The American Mathematical Monthly Vol. 79, No. 8 (Oct., 1972), pp. 928-941

### Classic Text Fills A Vacuum For Students

The study of the basic elements of smooth manifolds is one of the most important courses for mathematics and physics graduate students. Inexpensively priced, quality textbooks on the subject are currently particularly scarce. Matshushima’s book is a welcome addition to the literature in a very low priced edition.

Prerequisites for the course are solid undergraduate courses in real analysis of several variables, linear and abstract algebra and point-set topology. A previous classical differential geometry course on curve and surface theory isn’t really necessary, but will greatly enhance a first course in manifolds by supplying many low-dimensional examples in ℝn .

### Unique Textbook On Manifolds By A Master

The standard topics for such a course are all covered masterfully and concisely: Differentiable manifolds and their atlases,smooth mappings,immersions and embeddings, submanifolds, multilinear algebra, Lie groups and algebras, integration of differential forms and much more. This book is remarkable in it’s clarity and range, more so then most other introductions of the subject. Not only does it cover more material then most introductions to manifolds in a concise but readable manner, but it covers in detail several topics most introductions do not, such as homogeneous spaces and Lie subgroups.

Most significantly, it covers a major topic that most books at this level avoid: complex and almost complex manifolds. Complex and almost complex manifolds play important roles in both differential and algebraic geometry. They are also critical in the modern formulation of general relativity-particularly in modeling spacetime curvature near conditions of extreme gravitational force such as neutron stars and black holes. Despite these major roles in both pure mathematics and mathematical physics,almost all introductory textbooks on differentiable manifolds vehemently avoid both.

Part of the reason is the subject’s inherent difficulty. Complex manifolds requires sophisticated machinery such as sheaves and cohomology to describe once one gets beyond the essentials. Another reason is that complex manifolds are important in both differential geometry and its’ sister subject, algebraic geometry. As a result,it’s difficult sometimes to separate these aspects. By discussing only the barest essentials of complex manifolds,Mashushima avoids both these problems. This unique content makes the book far more valuable as a supplementary and reference text.

### Great Lost Classic Now Available Again For Graduate Students

Blue Collar Scholar is now proud to republish this lost classic in an inexpensive new edition. The book is suitable for strong undergraduates and first year graduate students of both mathematics and the physical sciences. BCS founder Karo Maestro has added his usual personal touch authoring a new preface. It introduces the student to smooth manifolds and gives Maestro’s recommended reading list for further study.

Matsushima’s book is a wonderful, self contained and inexpensive basis for a first course on the subject. It provides a strong foundation for either subsequent courses in differential geometry or advanced courses on smooth manifold theory.

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