Smooth Manifolds and Observables
Main themes of the book are manifolds, fibre bundles and differential operators acting on sections of vector bundles. A classical treatment of these topics starts with a coordinate description of a manifold M; the algebra of smooth functions on M is a derived object in this approach. The presented book is based on an alternative point of view, where calculus on manifolds is treated as a part of commutative algebra. In particular, the initial object is a commutative associative unital algebra F with certain additional properties. The corresponding smooth manifold is reconstructed as the spectrum of F. (A generalization to the non-commutative case, which is usually called non-commutative geometry, is based on this point of view. This generalization, however, is not treated in the book.) The first few chapters describe properties of algebras that correspond to smooth manifolds, introduce a notion of charts and atlases and define smooth maps between such manifolds. The authors (J. Nestruev is an invented name hiding a group of authors) then show equivalence of the algebraic definition with the usual one. In the second part of the book, the authors define tangent and cotangent fibre bundles of a manifold, jet bundles and they introduce linear differential operators in this algebraic setting. As explained throughout the book, and in particular in the appendix (written by A. M. Vinogradov), it is possible to give a motivation coming from classical mechanics for basic notions treated in the book. The commutative algebra F is related to the laboratory itself, elements of F to measuring devices and points in the spectrum of F to states of an observed physical system. The book contains quite a few exercises and many useful illustrations.