The study of the basic elements of smooth manifolds is one of the most important courses for mathematics and physics graduate students. Inexpensively priced and 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. The 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 . 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. Despite the fact complex and almost complex manifolds are incredibly important in both pure mathematics and mathematical physics-they play important roles in both differential and algebraic geometry, as well as in the modern formulation of geometry in general relativity, particularly in modeling spacetime curvature near conditions of extreme gravitational force such as neutron stars and black holes -almost all introductory textbooks on differentiable manifolds vehemently avoid both. Part of the reason is the subject's difficulty once one gets past the most basic elements, which is considerable and requires sophisticated machinery from algebra and topology such as sheaves and cohomology. Another reason is that complex manifolds are important in both differential geometry and its' sister subject, algebraic geometry-and 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 usually absent in introductory texts and presented by a master makes the book far more valuable as a supplementary and reference text. Blue Collar Scholar is now proud to republish this lost classic in an inexpensive new edition 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 with a preface introducing the student to smooth manifolds and a recommended reading list for further study. Matsushima's book is a wonderful, self contained and inexpensive basis for a first course on the subject that will provide a strong foundation for either subsequent courses in differential geometry or advanced courses on smooth manifold theory. This revised printing has several minor but important revisions in responding to several users' criticisms. Primary among these is significantly enlarged print-many readers complained about the tiny print size in the original. This has been rectified; the entire interior has been enlarged and carefully recropped. Also, several additional inexpensive references have been added to the Recommended Reading section. It is sincerely hoped these revisions will make this new printing of this edition of Matshushima's classic more user friendly and therefore an even better choice for an introductory graduate course in differential manifolds.