Titlebook: Smooth Manifolds and Observables; Jet Nestruev Textbook 2020Latest edition Springer Nature Switzerland AG 2020 Smooth manifolds.Smooth man

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Algebras and Points,r to the following fundamental question is presented: given an abstract .-algebra ., find a set (smooth manifold) . whose .-algebra of (smooth) functions can be identified with .. This involves giving the formal definition of the notion of . (which turns out to have a nontrivial structure!) and of t

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Smooth Manifolds (Algebraic Definition),nifold is given in this chapter. It is defined as the dual space . of any complete geometric algebra ., supplied with an open covering . in the Zariski topology such that each algebras . is isomorphic to .. Numerous concrete examples of how this works are presented.

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Charts and Atlases,ble manifold. The definition, of course, turns out to be equivalent to the algebraic one given in the previous chapter, as will be proved in Chapter .. Many concrete examples of the coordinate approach are presented.

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Smooth Maps,re available, the next natural step is to study smooth maps both in the classical case of differentiable manifolds and in their algebraic generalizations in the commutative algebra setting. This is done in the chapter. Numerous examples of general algebraic type, as well as of analytic coordinate ty

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Points, Spectra and Ghosts, algebras . over an arbitrary field . is considered. The .-. or . |.| of such algebras is defined just as in the case . and is supplied with the Zariski topology. The case of the field . (“Boolean algebras”) is studied separately. In the general case, the space |.| plays the role of manifold in this

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Differential Calculus as Part of Commutative Algebra,l geometry, say that of the derivative or tangent vector, are unsatisfactory, being of descriptive nature, and conceptually correct definitions are needed. The latter are provided by the differential calculus over commutative algebras. These conceptual definitions are of course equivalent to the cla

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Vector Bundles and Projective Modules,in the general algebraic situation. The notion of section of a vector bundle and its relationship with projective modules is described. The equivalence of the category of vector bundles over a manifold and the category of projective finite-type modules over the correspondence algebra of smooth funct

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Functors of the Differential Calculus and their Representations,ned notions are related by certain functors. These functors—the main functors of the differential calculus in commutative algebras—are called .. Some of them are studied here in detail, along with the corresponding objects and the natural transformations that represent them, as well as the main rule

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