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内容大纲
《广义微分几何》是该学科的第一本教科书，由美国数学协会出版，奠定了在理论物理中使用的微分几何主要领域的基础：可微性、卡坦微分学、同源和上同源、不同群、纤维束和连接等。书中还配有习题和解答有助于读者更好地学习。本书对研究微分几何或数学物理的学生与研究人员极为有用。 -
作者介绍
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目录
Preface to the revised print
Preface
Chapter 1. Diffeology and Diffeological Spaces
Linguistic Preliminaries
Euclidean spaces and domains
Sets, subsets, maps etc.
Parametrizations in sets
Smooth parametrizations in domains
Axioms of Diffeology
Diffeologies and diffeological spaces
Plots of a diffeological space
Diffeology or diffeological space?
The set of diffeologies of a set
Real domains as diffeological spaces
The wire diffeology
A diffeology for the circle
A diffeology for the square
A diffeology for the sets of smooth maps
Smooth Maps and the Category Diffeology
Smooth maps
Composition of smooth maps
Plots are smooth
Diffeomorphisms
Comparing Diffeologies
Fineness of diffeologies
Fineness via the identity map
Discrete diffeology
Coarse diffeology
Intersecting diffeologies
Infimum of a family of diffeologies
Supremum of a family of diffeologies
Playing with bounds
Pulling Back Diffeologies
Pullbacks of diffeologies
Smoothness and pullbacks
Composition of pullbacks
Inductions
What is an induction?
Composition of inductions
Criterion for being an induction
Surjective inductions
Subspaces of Diffeological Spaces
Subspaces and subset diffeology
Smooth maps to subspaces
Subspaces subsubspaces etc.
Inductions identify source and image
Restricting inductions to subspaces
Discrete subsets of a diffeological space
Sums of Diffeological Spaces
Building sums with spaces
Refining a sum
Foliated diffeology
Klein structure and singularities of a diffeological space
Pushing Forward Diffeologies
Pushforward of diffeologies
Smoothness and pushforwards
Composition of pushforwards
Subductions
What is a subduction?
Compositions of subductions
Criterion for being a subduction
Injective subductions
Quotients of Diffeological Spaces
Quotient and quotient diffeology
Smooth maps from quotients
Uniqueness of quotient
Sections of a quotient
Strict maps, between quotients and subspaces
Products of Diffeological Spaces
Building products with spaces
Projections on factors are subductions
Functional Diffeology
Functional diffeologies
Restriction of the functional diffeology
The composition is smooth
Functional diffeology and products
Functional diffeology on groups of diffeomorphisms
Slipping X into (X,X),X)
Functional diffeology of a diffeology
Iterating paths
Compact controlled diffeology
Generating Families
Generating diffeology
Generated by the empty family
Criterion of generation
Generating diffeology as projector
Fineness and generating families
Adding and intersecting families
Adding constants to generating family
Lifting smooth maps along generating families
Pushing forward families
Pulling back families
Nebula of a generating family
Dimension of Diffeological Spaces
Dimension of a diffeological space
The dimension is a diffeological invariant
Dimension of real domains
Dimension zero spaces are discrete
Dimensions and quotients of diffeologies
Chapter 2. Locality and Diffeologies
Chapter 3. Diffeological Vector Spaces
Chapter 4. Modeling Spaces, Manifolds, etc.
Chapter 5. Homotopy of Diffeological Spaces
Chapter 6. Cartan-De Rham Calculus
Chapter 7. Diffeological Groups
Chapter 8. Diffeological Fiber Bundles
Chapter 9. Symplectic Diffeology
Solutions to Exercises
Afterword
Notation and Vocabulary
Index
Bibliography
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