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Smooth Manifolds and Observables

  • Textbook
  • © 2020

Overview

Authors:
  1. Jet Nestruev

    1. Lizzano in Belvedere, Italy, Moscow, Russia

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  • Motivates the algebraic study of smooth manifolds by looking at them from the point of view of physics, in particular by using the observability principle
  • Bridges between ideas from physics, geometry, and algebra, offering new perspectives to students and researchers in each field
  • Incorporates a wide array of exercises and detailed illustrations, allowing readers to better familiarize themselves with the material
  • Includes ten new chapters in the second edition, as well as additional exercises, examples, and illustrations
  • Includes supplementary material: sn.pub/extras

Part of the book series: Graduate Texts in Mathematics (GTM, volume 220)

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Table of contents (21 chapters)

  1. Front Matter

    Pages i-xviii
    Download chapter PDF

  2. Introduction

    • Jet Nestruev
    Pages 1-11

  3. Cutoff and Other Special Smooth Functions on \({\mathbb R}^n\)

    • Jet Nestruev
    Pages 13-19

  4. Algebras and Points

    • Jet Nestruev
    Pages 21-36

  5. Smooth Manifolds (Algebraic Definition)

    • Jet Nestruev
    Pages 37-51

  6. Charts and Atlases

    • Jet Nestruev
    Pages 53-63

  7. Smooth Maps

    • Jet Nestruev
    Pages 65-76

  8. Equivalence of Coordinate and Algebraic Definitions

    • Jet Nestruev
    Pages 77-84

  9. Points, Spectra and Ghosts

    • Jet Nestruev
    Pages 85-100

  10. Differential Calculus as Part of Commutative Algebra

    • Jet Nestruev
    Pages 101-139

  11. Symbols and the Hamiltonian Formalism

    • Jet Nestruev
    Pages 141-150

  12. Smooth Bundles

    • Jet Nestruev
    Pages 151-170

  13. Vector Bundles and Projective Modules

    • Jet Nestruev
    Pages 171-202

  14. Localization

    • Jet Nestruev
    Pages 203-222

  15. Differential 1-forms and Jets

    • Jet Nestruev
    Pages 223-241

  16. Functors of the Differential Calculus and their Representations

    • Jet Nestruev
    Pages 243-272

  17. Cosymbols, Tensors, and Smoothness

    • Jet Nestruev
    Pages 273-292

  18. Spencer Complexes and Differential Forms

    • Jet Nestruev
    Pages 293-321

  19. The (Co)Chain Complexes Coming from the Spencer Sequence

    • Jet Nestruev
    Pages 323-340

  20. Differential Forms: Classical and Algebraic Approach

    • Jet Nestruev
    Pages 341-362
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Keywords

About this book

This textbook demonstrates how differential calculus, smooth manifolds, and commutative algebra constitute a unified whole, despite having arisen at different times and under different circumstances. Motivating this synthesis is the mathematical formalization of the process of observation from classical physics. A broad audience will appreciate this unique approach for the insight it gives into the underlying connections between geometry, physics, and commutative algebra.


The main objective of this book is to explain how differential calculus is a natural part of commutative algebra. This is achieved by studying the corresponding algebras of smooth functions that result in a general construction of the differential calculus on various categories of modules over the given commutative algebra. It is shown in detail that the ordinary differential calculus and differential geometry on smooth manifolds turns out to be precisely the particular case that corresponds to the category of geometric modules over smooth algebras. This approach opens the way to numerous applications, ranging from delicate questions of algebraic geometry to the theory of elementary particles.


Smooth Manifolds and Observables is intended for advanced undergraduates, graduate students, and researchers in mathematics and physics. This second edition adds ten new chapters to further develop the notion of differential calculus over commutative algebras, showing it to be a generalization of the differential calculus on smooth manifolds. Applications to diverse areas, such as symplectic manifolds, de Rham cohomology, and Poisson brackets are explored. Additional examples of the basic functors of the theory are presented alongside numerous new exercises, providing readers with many more opportunities to practice these concepts.

Authors and Affiliations

  • Lizzano in Belvedere, Italy, Moscow, Russia

    Jet Nestruev

About the author

Jet Nestruev is a collective of authors, who originally convened for a seminar run by Alexandre Vinogradov at the Mechanics and Mathematics Department of Moscow State University in 1969. In the present edition, Jet Nestruev consists of Alexander Astashov (Senior Researcher at the State Research Institute of Aviation Systems), Alexandre Vinogradov (Professor of Mathematics at Salerno University), Mikhail Vinogradov (Diffiety Institute), and Alexey Sossinsky (Professor at the Independent University of Moscow).

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