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**Introduction to differentiable manifolds.
2nd ed.**
*(English)*
Zbl 1008.57001

Universitext. New York, NY: Springer. xi, 250 p. (2002).

This volume – the second edition of the book on differential manifolds (the first edition 1962, Zbl 0103.15101) – represents an outgrowth of the author’s “Introduction to differentiable manifolds” (1962) and “Differential manifolds” (1972; Zbl 0239.58001). The book offers a quick introduction to basic concepts which are used in differential topology, differential geometry and differential equations. From the start the author assumes that all manifolds are finite dimensional. The book “Fundamentals of differential geometry”, Grad. Texts Math. 191 (1999; Zbl 0932.53001) can be viewed as a continuation of the present book (it covers the infinite dimensional case). This book consists of ten chapters; their contents are as follows:

Chapter I: Differential calculus (with proofs for the more important theorems). The author uses throughout the book the language of categories, the category of differentiable manifolds being of great importance. Chapter II: Manifolds. Chapter III: Vector bundles. In this part of the book the author develops purely formally certain functorial constructions having to do with vector bundles. Chapter IV: Vector fields and differential forms. Chapter V: Operations on vector fields and differential forms. These two chapters contain in detail the constructions associated with multilinear alternating forms and symmetric positive definite forms. Defining formally certain relations between functions, vector fields and differential forms, the author establishes a connection between the foundations of differential and Riemannian geometry. As an application he discusses the fundamental \(z\)-form and, in the next chapter, he connects it with Riemannian metrics in order to canonically construct the spray associated with such a metric. Chapter VI contains the proof of the theorem of Frobenius. In chapter VII: Metrics, the author completes the uniqueness theorem on tubular neighborhoods by showing that, when a Riemannian metric is given, a tubular neighborhood can be straightened out to a metric one. Then, he shows how a Riemannian metric gives rise, in a natural way, to a spray and thus how one recovers geodesics. The material of chapter VIII ‘Integration of differential forms’ is also contained in his book “Real and functional analysis”, Grad. Texts Math. 142 (1993; Zbl 0831.46001) and is based only on standard properties of integration in Euclidean space. Chapter IX ‘Stokes’s theorem’ and Chapter X ‘Applications of Stokes’s theorem’ contain the proofs of Stokes’s theorem for a rectangular simplex on a manifold and the case with singularities. This last chapter is a survey of the applications of Stokes’s theorem: the computation of the maximal de Rham cohomology, the computation of the volume form and the proofs for the divergence theorem, Cauchy’s theorem and the Poincaré residue theorem (using complex manifolds).

The bibliography contains important new titles in studying differential geometry. A large index is also included.

This is an interesting Universitext (for students – the first year graduate level or advanced undergraduate level), with important concepts concerning the general basic theory of differential manifolds.

Chapter I: Differential calculus (with proofs for the more important theorems). The author uses throughout the book the language of categories, the category of differentiable manifolds being of great importance. Chapter II: Manifolds. Chapter III: Vector bundles. In this part of the book the author develops purely formally certain functorial constructions having to do with vector bundles. Chapter IV: Vector fields and differential forms. Chapter V: Operations on vector fields and differential forms. These two chapters contain in detail the constructions associated with multilinear alternating forms and symmetric positive definite forms. Defining formally certain relations between functions, vector fields and differential forms, the author establishes a connection between the foundations of differential and Riemannian geometry. As an application he discusses the fundamental \(z\)-form and, in the next chapter, he connects it with Riemannian metrics in order to canonically construct the spray associated with such a metric. Chapter VI contains the proof of the theorem of Frobenius. In chapter VII: Metrics, the author completes the uniqueness theorem on tubular neighborhoods by showing that, when a Riemannian metric is given, a tubular neighborhood can be straightened out to a metric one. Then, he shows how a Riemannian metric gives rise, in a natural way, to a spray and thus how one recovers geodesics. The material of chapter VIII ‘Integration of differential forms’ is also contained in his book “Real and functional analysis”, Grad. Texts Math. 142 (1993; Zbl 0831.46001) and is based only on standard properties of integration in Euclidean space. Chapter IX ‘Stokes’s theorem’ and Chapter X ‘Applications of Stokes’s theorem’ contain the proofs of Stokes’s theorem for a rectangular simplex on a manifold and the case with singularities. This last chapter is a survey of the applications of Stokes’s theorem: the computation of the maximal de Rham cohomology, the computation of the volume form and the proofs for the divergence theorem, Cauchy’s theorem and the Poincaré residue theorem (using complex manifolds).

The bibliography contains important new titles in studying differential geometry. A large index is also included.

This is an interesting Universitext (for students – the first year graduate level or advanced undergraduate level), with important concepts concerning the general basic theory of differential manifolds.

Reviewer: Corina Mohorianu (Iaşi)