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Geometry of differential forms. Transl. from the Japanese by Teruko Nagase and Katsumi Nomizu. (English) Zbl 0987.58002
Translations of Mathematical Monographs. Iwanami Series in Modern Mathematics. 201. Providence, RI: American Mathematical Society (AMS). xxiv, 321 p. (2001).
This book is a translation of the original version, which was published in two volumes, in Japanese. It essentially covers virtually everything that one would want to see in a course in modern differential geometry.
Chapter 1 begins with the definition of a differentiable manifold and basic associated notions such as tangent vectors and vector fields. Chapter 2 introduces differential forms, defines fundamental operations (exterior product, exterior differentiation, Lie derivative) and gives a proof of the Frobenius theorem. The de Rham theorem is the subject of Chapter 3, which covers de Rham cohomology and applications of the de Rham theorem. Chapter 4 develops the relationship between differential forms and Riemannian metrics. Hodge theory and its applications (Poincaré duality and Euler number) are well covered. The notions of vector bundles and characteristic classes, which are central to modern differential geometry, are developed in Chapter 5. The concepts of connection and curvature in vector bundles are explored here, as well as Pontrjagin classes, Chern classes, Euler classes, and the Gauss-Bonnet theorem.
The last chapter, 6, explains the theory of characteristic classes, which in the words of the author, is truly a summit in modern geometry.
The author concludes with some interesting perspectives on the role played by the modern theory in today’s geometry. The stated goal of the author of providing the reader with the flavor of modern geometry has indeed been accomplished.

58A10 Differential forms in global analysis
58-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to global analysis
58A05 Differentiable manifolds, foundations
58A15 Exterior differential systems (Cartan theory)
58A12 de Rham theory in global analysis