- #1
- 22,183
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- Author: Michael Spivak
- Title: A Comprehensive Introduction to Differential Geometry
- Amazon Link:
https://www.amazon.com/dp/0914098705/?tag=pfamazon01-20
https://www.amazon.com/dp/0914098713/?tag=pfamazon01-20
https://www.amazon.com/dp/0914098721/?tag=pfamazon01-20
https://www.amazon.com/dp/091409873X/?tag=pfamazon01-20
https://www.amazon.com/dp/0914098748/?tag=pfamazon01-20 - Prerequisities: Calculus on Manifolds (a rigorous calculus 1,2,3 course), Linear Algebra
- Level: Undergrad
Table of Contents for Volume I:
Code:
[LIST]
[*] Manifolds
[LIST]
[*] Elementary properties of manifolds
[*] Examples of manifolds
[*] Problems
[/LIST]
[*] Differential Structures
[LIST]
[*] C^\infty structures
[*] C^\infty functions
[*] Partial derivatives
[*] Critical points
[*] Immersion theorems
[*] Partitions of unity
[*] Problems
[/LIST]
[*] The Tangent Bundle
[LIST]
[*] The tangent space of R^n
[*] The tangent space of an imbedded manifold
[*] Vector bundles
[*] The tangent bundle of a manifold
[*] Equivalence classes of curves, and derivations
[*] Vector fields
[*] Orientation
[*] Addendum. Equivalence of Tangent Bundles
[*] Problems
[/LIST]
[*] Tensors
[LIST]
[*] The dual bundle
[*] The differential of a function
[*] Classical versus modern terminology
[*] Multilinear functions
[*] Covariant and contravariant tensors
[*] Mixed tensors, and contraction
[*] Problems
[/LIST]
[*] Vector Fields and Differential Equations
[LIST]
[*] Integral curves
[*] Existence and uniqueness theorems
[*] The local flow
[*] One-parameter groups of diffeomorphisms
[*] Lie derivatives
[*] Brackets
[*] Addendum: Differential Equations
[*] Addendum: Parameter Curves in Two Dimensions
[*] Problems
[/LIST]
[*] Integral Manifolds
[LIST]
[*] Prologue; classical integrability theorems
[*] Local Theory; Frobenius integrability theorem
[*] Global Theory
[*] Problems
[/LIST]
[*] Differential Forms
[LIST]
[*] Alternating functions
[*] The wedge product
[*] Forms
[*] Differential of a form
[*] Frobenius integrability theorem (second version)
[*] Closed and exact forms
[*] The Poincare Lemma
[*] Problems
[/LIST]
[*] Integration
[LIST]
[*] Classical line and surface integrals
[*] Integrals over singular k-cubes
[*] The boundary of a chain
[*] Stokes' Theorem
[*] Integrals over manifolds
[*] Volume elements
[*] Stokes' Theorem
[*] de Rham cohomology
[*] Problems
[/LIST]
[*] Riemannian Metrics
[LIST]
[*] Inner products
[*] Riemannian metrics
[*] Length of curves
[*] The calculus of variations
[*] The First Variation Formula and geodesics
[*] The exponential map
[*] Geodesic completeness
[*] Addendum: Tubular Neighborhoods
[*] Problems
[/LIST]
[*] Lie Groups
[LIST]
[*] Lie groups
[*] Left invariant vector fields
[*] Lie algebras
[*] Subgroups and subalgebras
[*] Homomorphisms
[*] One-parameter subgroups
[*] The exponential map
[*] Closed subgroups
[*] Left invariant forms
[*] Bi-invariant metrics
[*] The equations of structure
[*] Problems
[/LIST]
[*] Excursion in the Real of Algebraic Topology
[LIST]
[*] Complexes and exact sequences
[*] The Mayer-Vietoris sequence
[*] Triangulations
[*] The Euler characteristic
[*] Mayer-Vietoris sequence for compact supports
[*] The exact sequence of a pair
[*] Poincare Duality
[*] The Thorn class
[*] Index of a vector field
[*] Poincare-Hopf Theorem
[*] Problems
[/LIST]
[*] Appendix
[LIST]
[*] To Chapter 1
[*] Problems
[*] To Chapter 2
[*] Problems
[*] To Chapter 6
[*] To Chapters 7,
[/LIST]
[*] Notation Index
[*] Index
[/LIST]
Code:
[LIST]
[*] Curves in the Plane and in Space
[LIST]
[*] Curvature of plane curves
[*] Convex curves
[*] Curvature and torsion of space curves
[*] The Serret-Frenet formulas
[*] The natural form on a Lie group
[*] Classification of plane curves under the group of special affine motions
[*] Classification of curves in R^n
[/LIST]
[*] What they knew about Surfaces before Gauss
[LIST]
[*] Euler's Theorem
[*] Meusnier's Theorem
[/LIST]
[*] The Curvature of Surfaces in Space
[LIST]
[*] How to Read Gauss
[*] Gauss' Theory of Surfaces
[LIST]
[*] The Gauss map
[*] Gaussian curvature
[*] The Weingarten map; the first and second fundamental forms
[*] The Theorema Egregium
[*] Geodesics on a surface
[*] The metric in geodesic polar coordinates
[*] The integral of the curvature over a geodesic triangle
[*] Addendum. The formula of Bertrand and Puiseux; Diquet's formula
[/LIST]
[/LIST]
[*] The Curvature of Higher Dimensional Manifolds
[LIST]
[*] An Inaugural Lecture
[LIST]
[*] "On the Hypotheses which lie at the Foundations of Geometry"
[/LIST]
[*] What Did Riemann Say?
[LIST]
[*] The form of the metric in Riemannian normal coordinates
[/LIST]
[*] A Prize Essay
[*] The Birth of the Riemann Curvature Tensor
[LIST]
[*] Necessary conditions for a metric to be flat
[*] The Riemann curvature tensor
[*] Sectional Curvature
[*] The Test Case; first version
[*] Addendum. Finsler metrics
[/LIST]
[/LIST]
[*] The Absolute Differential Calculus (The Ricci Calculus); Or, the Debauch of Indices
[LIST]
[*] Covariant derivatives
[*] Ricci's Lemma
[*] Ricci's identities
[*] The curVature tensor
[*] The Test Case; second version
[*] Classical connections
[*] The torsion tensor
[*] Geodesics
[*] Bianchi's identities
[/LIST]
[*] The Nabla Operator
[LIST]
[*] Kozul connections
[*] Covariant derivatives
[*] Parallel translation
[*] The torsion tensor
[*] The Levi-Civita connection
[*] The curvature tensor
[*] The Test Case; third version
[*] Bianchi's identities
[*] Geodesics
[*] The First Variation Formula
[*] Addendum: Connections with the same geodesics
[*] Addendum: Riemann's invariant definition of the curvature tensor
[/LIST]
[*] The Repere Mobile (the Moving Frame)
[LIST]
[*] Moving frames
[*] The structural equations of Euclidean space
[*] The Structural equations of a Riemannian manifold
[*] The Test Case; fourth version
[*] Adapted frames
[*] The structural equations in polar coordinates
[*] The Test Case; fifth version
[*] The Test Case; sixth version
[*] "The curvature determines the metric"
[*] The 2-dimensional case
[*] Cartan connections
[*] Covariant derivatives and the torsion and curvature tensors
[*] Bianchi's identities
[*] Addendum: Manifolds of constant curvature
[LIST]
[*] Schur's Theorem
[*] The form of the metric in normal coordinates
[/LIST]
[*] Addendum: Conformally equivalent manifolds
[*] Addendum: E. Cartan's treatment of normal coordinates
[/LIST]
[*] Connections in Principal Bundles
[LIST]
[*] Principal bundles
[*] Lie groups acting on manifolds
[*] A new definition of Cartan connections
[*] Ehresmann connections
[*] Lifts
[*] Parallel translation and covariant derivatives
[*] The covariant differential and the curvature form
[*] The dual form and the torsion form
[*] The structural equations
[*] The torsion and curvature tensors
[*] The Test Case; seventh version
[*] Bianchi's identities
[*] Summary
[*] Addendum: The tangent bundle of F(M)
[*] Addendum: Complete connections
[*] Addendum: Connections in vector bundles
[*] Addendum: Flat connections
[/LIST]
[*] Notation Index
[*] Index
[/LIST]
Table of Contents for Volume III:
Code:
[LIST]
[*] The Fundamental Equations for Hypersurfaces
[LIST]
[*] Covariant differentiation in a submanifold of a Riemannian manifold
[*] The second fundamental form, the Gauss formulas, and Gauss' equation; Synge's inequality
[*] The Weingarten equations and the Codazzi-Mainardi equations for hypersurfaces
[*] The classical tensor analysis description
[*] The moving frame description
[*] Addendum. Auto-parallel and totally geodesic submanifolds
[*] Problems
[/LIST]
[*] Elements of the Theory of Surfaces
[LIST]
[*] The first and second fundamental forms
[*] Classification of points on a surface; the osculating paraboloid and the Dubin indicatrix
[*] Principal directions and curvatures, asymptotic directions, flat points and umbilics; all-umbilic surfaces
[*] The classical Gauss formulas, Weingarten equations, Gauss equation, and Codazzi-Mainardi equations
[*] Fundamental theorem of surface theory
[*] The third fundamental form
[*] Convex surfaces; Hadamard's theorem
[*] The fundamental equations via moving frames
[*] Review of Lie groups
[*] Application of Lie groups to surface theory; the fundamental equations and the structural equations of SO(3)
[*] Affine surface theory; the osculating paraboloids and the affine invariant conformal structure
[*] The special affine first fundamental form
[*] Quadratic and cubic forms; apolarity
[*] The affine normal direction; the special affine normal
[*] The special affine Gauss formulas and special affine second fundamental form
[*] The Pick invariant; surfaces with Pick invariant 0
[*] The special affine Weingarten formulas
[*] The special affine Codazzi-Mainardi equations; the fundamental theorem of special affine surface theory
[*] Problems
[/LIST]
[*] A Compendium of Surfaces
[LIST]
[*] Basic calculations
[*] The classical flat surfaces
[*] Ruled surfaces
[*] Quadric surfaces
[*] Surfaces of revolution
[LIST]
[*] Rotation surfaces of constant curvature
[/LIST]
[*] Minimal surfaces
[*] Addendum. Envelopes of 1-parameter families of planes
[*] Problems
[/LIST]
[*] Curves on Surfaces
[LIST]
[*] Normal and geodesic curvature
[*] The Darboux frame; geodesic torsion
[*] Laguerre's theorem
[*] General properties of lines of curvature, asymptotic curves, and geodesics
[*] The Beltrami-Enneper theorem
[*] Lines of curvature and Dupin's theorem
[*] Conformal maps of R^3; Liouville's theorem
[*] Geodesies and Clairaut's theorem
[*] Addendum: Special parameter curves
[*] Addendum: Singularities of line fields
[*] Problems
[/LIST]
[*] Complete Surface of Constant Curvature
[LIST]
[*] Hilbert's lemma; complete surfaces of constant curvature K > 0
[*] Analysis of flat surfaces; the classical classification of developable surfaces
[*] Complete flat surfaces
[*] Complete surfaces of constant curvature K < 0
[/LIST]
[*] The Gauss-Bonnet Theorem and Related Topics
[LIST]
[*] The connection form for an orthornormal moving frame on a surface; the change in angle under parallel translation
[*] The integral of K dA over a polygonal region
[*] The Gauss-Bonnet theorem; consequences
[*] Total absolute curvature of surfaces
[*] Surfaces of minimal total absolute curvature
[*] Total curvature of curves; Fenchel's theorem, and the Fary-Milnor theorem
[*] Addendum: Compact surfaces with constant negative curvature
[*] Addendum: The degree of the normal map
[*] Problems
[/LIST]
[*] Mini-Bibliography for Volume III
[*] Notation Index
[*] Index
[/LIST]
Table of Contents for Volume IV:
Code:
[LIST]
[*] Higher Dimensions and Codimensions
[LIST]
[*] The Geometry of Constant Curvature Manifolds
[LIST]
[*] The standard models of S^n(K_0) and H^n(K_0) in R^{n+1}
[*] Stereographic projection and the conformal model of H^n
[*] Conformal maps of R^n and the isometries of H^n
[*] Totally geodesic submanifolds and geodesic spheres of H^n
[*] Horospheres and equidistant hypersurfaces
[*] Geodesic mappings; the projective model of H^n; Beltrami's theorem
[/LIST]
[*] Curves in a Riemannian Manifold
[LIST]
[*] Frenet frames and curvatures
[*] Curves whose jth curvature vanish
[/LIST]
[*] The Fundamental Equations for Submanifolds
[LIST]
[*] The normal connection and the Weingarten equations
[*] Second fundamental forms and normal fundamental forms; the Codazzi-Mainardi equations
[*] The Ricci equations
[*] The fundamental theorem for submanifolds of Euclidean space
[*] The fundamental theorem for submanifolds of constant curvature manifolds
[/LIST]
[*] First Consequences
[LIST]
[*] The curvatures of a hypersurface; Theorema Egregium; formula for the Gaussian curvature
[*] The mean curvature normal; umbilics; all-umbilic submanifolds of Euclidean space
[*] All-umbilic submanifolds of constant curvature manifolds
[*] Positive curvature and convexity
[/LIST]
[*] Further Results
[LIST]
[*] Flat ruled surfaces in R^m
[*] Flat ruled surfaces in constant curvature manifolds
[*] Curves on hypersurfaces
[/LIST]
[*] Complete Surfaces of Constant Curvature
[LIST]
[*] Modifications of results for surfaces in R^3
[*] Surfaces of constant curvature in S^3
[LIST]
[*] surfaces with constant curvature 0
[*] the Hopf map
[/LIST]
[*] Surfaces of constant curvature in H^3
[LIST]
[*] Jorgens theorem; surfaces of constant curvature 0
[*] surfaces of constant curvature — 1
[*] rotation surfaces of constant curvature between — 1 and 0
[/LIST]
[/LIST]
[*] Hypersurfaces of Constant Curvature in Higher Dimensions
[LIST]
[*] Hypersurfaces of constant curvature in dimensions >3
[*] The Ricci tensor; Einstein spaces, hypersurfaces which are Einstein spaces
[*] Hypersurfaces of the same constant curvature as the ambient manifold
[*] Addendum: The Laplacian
[*] Addendum: The * operator and the Laplacian on forms; Hodge's Theorem
[*] Addendum: When are two Riemannian manifolds isometric?
[*] Addendum: Better imbedding invariants
[*] Problems
[/LIST]
[/LIST]
[*] The Second Variation
[LIST]
[*] Two-parameter variations; the second variation formula
[*] Jacobi fields; conjugate points
[*] Minimizing and non-minimizing geodesies
[*] The Hadamard-Cartan Theorem
[*] The Sturm Comparison Theorem; Bonnet's Theorem
[*] Generalizations to higher dimensions; the Morse-Schoenberg Comparison Theorem; Meyer's Theorem; the Rauch Comparison Theorem
[*] Synge's lemma; Synge's Theorem
[*] Cut points; Klingenberg's theorem
[*] Problems
[/LIST]
[*] Variations of Length, Area, and Volume
[LIST]
[*] Variation of area for normal variations of surfaces in R^3; minimal surfaces
[*] Isothermal coordinates on minimal surfaces; Bernstein's Theorem
[*] Weierstrass-Enneper representation
[*] Associated minimal surfaces; Schwarz's Theorem
[*] Change of orientation; Henneberg's minimal surface
[*] Classical calculus of variations in n dimensions
[*] Variation of volume formula
[*] Isoperimetric problems
[*] Addendum: Isothermal coordinates
[*] Addendum: Immersed spheres with constant mean curvature
[*] Addendum: Imbedded surfaces with constant mean curvature
[*] Addendum: The second variation of volume
[/LIST]
[*] Mini-Bibliography for Volume IV
[*] Notation Index
[*] Index
[/LIST]
Table of Contents for Volume V:
Code:
[LIST]
[*] And now a Brief Message from our Sponsor
[LIST]
[*] First Order PDE's
[LIST]
[*] Linear first order PDE's; characteristic curves; Cauchy problem for free initial curves
[*] Quasi-linear first order PDE's; characteristic curves; Cauchy problem lor free initial conditions; characteristic initial conditions
[*] General first order PDE's; Monge cone; characteristic curves of a solution; characteristic strips; Cauchy problem for free initial data; characteristic initial data
[*] First order PDE's in n variables
[/LIST]
[*] Free Initial Manifolds for Higher Order Equations
[*] Systems of First Order PDE's
[*] The Cauchy-Kowalewski Theorem
[*] Classification of Second Order PDE's
[LIST]
[*] Classification of semi-linear equations
[*] Reduction to normal forms
[*] Classification of general second order equations
[/LIST]
[*] The Prototypical PDE's of Physics
[LIST]
[*] The wave equation; the heat equation; Laplace's equation
[*] Elementary properties
[/LIST]
[*] Hyperbolic Systems in Two Variables
[*] Hyperbolic Second Order Equations in Two Variables
[LIST]
[*] First reduction of the problem
[*] New system of characteristic equations
[*] Characteristic initial data
[*] Monge-Ampere equations
[/LIST]
9. Elliptic Solutions of Second Order Equations in Two Variables
[LIST]
[*] Addendum: Differential systems; the Cartan-Kahler Theorem
[*] Addendum: An elementary maximum principal
[*] Problem
[/LIST]
[/LIST]
[*] Existence and Non-Existence of Isometric Imbeddings
[LIST]
[*] Non-imbeddability theorems; exteriorly orthogonal bilinear forms; index of nullity and index of relative nullity
[*] The Darboux equation
[*] Burstin-Janet-Cartan Theorem
[*] Addendum. The embedding problem via differential systems
[*] Problems
[/LIST]
[*] Rigidity
[LIST]
[*] Rigidity in higher dimensions; type number
[*] Bendings, warpings, and infinitesimal bendings
[*] R^3 -valued differential forms, the support function, and Minkowski's formulas
[*] Infinitesimal rigidity of convex surfaces
[*] Cohn-Vossen's Theorem
[*] Minkowski's Theorem
[*] Christoffel's Theorem
[*] Other problems, solved and unsolved
[*] Local problems; the role of the asymptotic curves
[*] Other classical results
[*] E. E. Levi's Theorems and Schilt's Theorem
[*] Surfaces in S^3 and H^3
[*] Rigidity for higher codimension
[*] Addendum. Infinitesimal bendings of rotation surfaces
[*] Problems
[/LIST]
[*] The Generalized Gauss-Bonnet Theorem and What it Means for Mankind
[LIST]
[*] Historical remarks
[*] Operations on Bundles
[LIST]
[*] Bundle maps and principal bundle maps; Whitney sums and induced bundles; the covering homotopy theorem
[/LIST]
[*] Grassmannians and Universal Bundles
[*] The Pfaffian
[*] Defining the Euler class in Terms of a Connection
[LIST]
[*] The Euler class
[*] The class C(\xi)
[*] The Gauss-Bonnet-Chern Theorem
[/LIST]
[*] The Concept of Characteristic Classes
[*] The Cohomology of Homogeneous Spaces
[LIST]
[*] The C^\infty structure of homogeneous spaces
[*] Invariant forms
[/LIST]
[*] A Smattering of Classical Invariant Theory
[LIST]
[*] The Capelli identities
[*] The first fundamental theorem of invariant theory for O(n) and SO(n)
[/LIST]
[*] An Easier Invariance Problem
[*] The Cohomology of the Oriented Grassmannians
[LIST]
[*] Computation of the cohomology; Pontryagin classes
[*] Describing the characteristic classes in terms of a connection
[/LIST]
[*] The Weil Homomorphism
[*] Complex Bundles
[LIST]
[*] Hermitian inner products, the unitary group, and complex Grassmanians
[*] The cohomology of the complex Grassmanians; Chern classes
[*] Relations between the Chern classes and the Pontryagin and Euler classes
[/LIST]
[*] Valedictory
[LIST]
[*] Addendum: Invariant theory for the unitary group
[*] Addendum: Recovering the differential forms; the Gauss-Bonnet-Chern Theorem for manifolds-with-boundary
[/LIST]
[/LIST]
[*] Bibliography
[LIST]
[*] Other topics in Differential Geometry
[*] Books
[*] Journal articles
[/LIST]
[*] Notation Index
[*] Index
[/LIST]
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