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**Smooth manifolds and observables.**
*(English)*
Zbl 1021.58001

Graduate Texts in Mathematics. 220. New York, NY: Springer. xiv, 222 p. (2003).

Jet Nestruev is a fictitious name for a group of authors, led by Alexandre Vinogradov and his ‘diffiety group’.

The book provides a self-contained introduction to the theory of smooth manifolds and fibre bundles, oriented towards graduate students in mathematics and physics. The approach followed here, however, substantially differs from most textbooks on manifold theory. The starting element is a commutative \(\mathbf R\)-algebra and points of the manifold on which this is to become the algebra of smooth functions are the algebra homomorphisms into \(\mathbf R\) (which have a bijective correspondence to the prime ideals of the algebra). Such an approach requires rather more sophisticated tools of mathematics, but it is strongly motivated by arguments taken from physics. Fundamental notions in theoretical physics, such as ‘observables’, the ‘state of a physical system’ and ‘measuring devices’ are called upon throughout the book to justify this algebro-geometric approach. The general philosophy is that also in mathematics there should be a concern about ‘observability’.

After some introductory material in the first two chapters, chapter 3 embarks on the mathematical exposition of the algebraic approach. It forms the basis for further developments. Of particular interest in that respect is the concept of a complete, geometric \(\mathbf R\)-algebra. In chapter 4, the algebraic definition of a smooth manifold is studied, together with basic operations and concepts such as product manifold, mapping between manifolds, submanifolds, etc. The standard approach to manifold theory, using local charts and atlases, is briefly explained in chapter 5, its equivalence to the algebraic approach being the subject of chapter 7. Chapter 8 is an excursion into more general commutative algebras, leading to a discussion of the (prime) spectrum of an algebra as the set of points of the associated manifold and to the definition of ghost ideals. Chapters 9 to 11 are about the differential calculus, tangent and cotangent bundles, differential operators and manifolds of jets, arbitrary smooth bundles and vector bundles, always of course in the spirit of the book, i.e. with the emphasis on the algebra of functions.

This book is certainly quite interesting and may appeal even to people who merely want to study algebraic geometry, in the sense that they will gain extra insight here by the attention which is paid to making certain constructions in algebraic geometry physically or intuitively acceptable. The ‘Afterword’ of the book promises us an ‘infinite series’ of forthcoming ones.

The book provides a self-contained introduction to the theory of smooth manifolds and fibre bundles, oriented towards graduate students in mathematics and physics. The approach followed here, however, substantially differs from most textbooks on manifold theory. The starting element is a commutative \(\mathbf R\)-algebra and points of the manifold on which this is to become the algebra of smooth functions are the algebra homomorphisms into \(\mathbf R\) (which have a bijective correspondence to the prime ideals of the algebra). Such an approach requires rather more sophisticated tools of mathematics, but it is strongly motivated by arguments taken from physics. Fundamental notions in theoretical physics, such as ‘observables’, the ‘state of a physical system’ and ‘measuring devices’ are called upon throughout the book to justify this algebro-geometric approach. The general philosophy is that also in mathematics there should be a concern about ‘observability’.

After some introductory material in the first two chapters, chapter 3 embarks on the mathematical exposition of the algebraic approach. It forms the basis for further developments. Of particular interest in that respect is the concept of a complete, geometric \(\mathbf R\)-algebra. In chapter 4, the algebraic definition of a smooth manifold is studied, together with basic operations and concepts such as product manifold, mapping between manifolds, submanifolds, etc. The standard approach to manifold theory, using local charts and atlases, is briefly explained in chapter 5, its equivalence to the algebraic approach being the subject of chapter 7. Chapter 8 is an excursion into more general commutative algebras, leading to a discussion of the (prime) spectrum of an algebra as the set of points of the associated manifold and to the definition of ghost ideals. Chapters 9 to 11 are about the differential calculus, tangent and cotangent bundles, differential operators and manifolds of jets, arbitrary smooth bundles and vector bundles, always of course in the spirit of the book, i.e. with the emphasis on the algebra of functions.

This book is certainly quite interesting and may appeal even to people who merely want to study algebraic geometry, in the sense that they will gain extra insight here by the attention which is paid to making certain constructions in algebraic geometry physically or intuitively acceptable. The ‘Afterword’ of the book promises us an ‘infinite series’ of forthcoming ones.

Reviewer: Willy Sarlet (Gent)