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Geometry A Comprehensive Introduction to Differential Geometry series by Spivak

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  1. Jan 24, 2013 #1

    micromass

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    Table of Contents for Volume I:
    Code (Text):

    [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]
     

    Table of Contents for Volume II
    Code (Text):

    [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 (Text):

    [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 (Text):

    [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 (Text):

    [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]
     
     
    Last edited by a moderator: May 6, 2017
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  3. Jan 24, 2013 #2

    mathwonk

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    this is a fabulous book. Mike taught the course at Brandeis one year and began it there and then could not stop writing for years afterwards. The first volume has little geometry but is a great introduction to differential manifolds and lie groups and a little differential topology including de Rham's theorem.

    Then the superb part is volume 2, reproducing and translating the great works of Riemann and Gauss on curvature and explaining them in modern language. This is the place to learn about curvature, the central idea of differential geometry. Great book!

    much later (November 2015) I am reading chapter 5 vol I and enjoying it a lot. I am finally learning some of this stuff it feels like for the first time. Probably I only skimmed this book before. I really like the geometric concept of a one parameter group of diffeomeorphisms determined by a smooth vector field on a compact manifold as he explains it. I also see now that he took much of this from Lang's book Analysis II, but clarified it and added details and filled gaps in Lang's usual free flowing style of exposition. Lang did a nice job too of showing how the implicit function theorem makes the proof of local smoothness of the flow look easy. I would sketch it here but last time i tried the browser trashed my whole edit.

    I am a little embarrassed that I praised it so much without really having read it in detail. But the more I read the more I feel justified in my praise. At last I am learning what a Lie derivative is, (with respect to a vector field), and why it can be defined for basically anything. I.e. a vector field defines a one parameter family of diffeomorphisms {ft} by flowing along the vector field for time t. Then since a diffeomorphism lets you pull back virtually anything, you can subtract the values of your object at time t from the value at time 0 and take a limit. Wow! cool.

    In general Spivak seems very conversational and explains well what is going on. He does not just present the material he explains why we want it and what it means.

    Now that I am retired I have time it seems to actually read books and learn stuff that will not be needed for my vita update every year.
     
    Last edited: Nov 8, 2015
  4. Jan 24, 2013 #3

    WannabeNewton

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    mathwonk how would you rank the difficulty of the first vol. in relation to Lee's smooth manifolds text?
     
  5. Jan 24, 2013 #4

    mathwonk

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    well i have not read lee, who is very well liked. i am guessing that lee learned from spivak, then spent years writing his own version. i would look at both. everyone says lee is excellent. I do recall Spivak made the construction of the tangent bundle look kind of long and tedious. In fact he says himself his class had to take a week long "rest cure" after that chapter, and after some others.

    Ok I just read some parts of Lee, specifically the section on tangent vectors. I can see how it would be very helpful and clear to a beginner, but as someone who already knows the material it seems overly wordy and tedious to wade through. I recommend it to someone learning it for the first time, but afterwards it is also helps to have books that summarize the main points briefly and succinctly, so you can remember them and focus on what matters most.

    I also happen to disagree with his choice to define a tangent vector as a derivation on global functions, even if that has many technical advantages; but it is not a geometrical definition hence not intuitive to me. Lee does an excellent job of motivating it though and explaining why he chose it. I do not agree either with his argument that using local derivations is needlessly abstract, since germs are not really that far out. The only really intuitive definition is the equivalence relation on curves in my opinion and he rejects it mainly because it takes some work to show it has a vector space structure. I also think it would be very useful as an exercise for the reader to work out some of the other equivalent definitions, or if Lee had done one of them so the reader could compare and see which he likes best.

    As a remark, I think he is slightly inaccurate to assert that germs are necessary for the derivation definition in the analytic and algebraic cases. I.e. although global analytic and algebraic functions do not always exist say on compact or projective manifolds, they do exist on stein or affine neighborhoods, so one can avoid the germ concept by using the point derivations on functions defined say on an affine neighborhood. This is essentially the same as the embedding definition, or carrying over the tangent space from a embedded model or coordinate model, and I think, if explained, would thus help the student grasp the abstract version used in the book.

    This is a conundrum, how to explain things clearly without going on at length in a way that makes the reading tiresome. But I think Lee has done a service here by making the subject seem quite reasonable to learn on ones own. So I see why students like it, but I still prefer Spivak for myself.
     
    Last edited: Nov 8, 2015
  6. Jan 24, 2013 #5

    WannabeNewton

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    Oh I already have lee's books but I was just wondering if spivak happened to go more advanced in certain topics (particularly differential forms/exterior calculus and lie groups)
     
  7. Jan 26, 2013 #6

    mathwonk

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    how advanced can you get about diff forms? an n - diml smooth manifold has at each point a tangent vector space of dim n, hence a space of alternating k-linear functionals of dim "n choose k". For each k, these spaces fit together into a vector bundle, sections of which are called k - forms.

    there is a linear map d called exterior differentiation from k forms to k+1 forms. The image of d is called the space of exact forms and the kernel is called the space of closed forms. The deRham theorem says that, on a compact smooth manifold, the space of closed k forms modulo the space of exact k forms is isomorphic to the singular (and Cech) k dimensional cohomology space. The isomorphism is given by integration over homology cycles.

    That's the basics.
     
  8. Jan 26, 2013 #7

    dextercioby

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    Mathwonk, how would you rate Spivak's books against Kobayashi and Nomizu's 2 volume text ?
     
  9. Jan 26, 2013 #8

    WannabeNewton

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    I may not have been clear in my question; I meant in terms of problem sets.
     
  10. Jan 26, 2013 #9

    mathwonk

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    i have not read kobayashi and nomizu, but that itself says something because i tried. i recall it was dry dry dry, as opposed to spivak which is readable and fun. this is not a mathematical criticism of k and n, just an admission that i could not manage it. try it yourself. maybe you have more tenacity.
     
  11. Feb 2, 2013 #10
    Spivak really loves differential geometry, as these books show (I will restrict myself to the first two volumes, for I am unfamiliar with the rest). Everything is motivated with the utmost care--all the abstract topological stuff in the first volume is made completely natural in setting up the geometric content of the second volume. The translated papers of Gauss and Riemann are a huge plus.
     
  12. Sep 20, 2013 #11
    Hi,
    I've just bought a bundle of vol I and vol II in USA ( I'm in Italy ) which was very low priced. This is actually a second edition. From your experience, are there a lot of differences between second and third edition? The google books preview or amazon one seems to spoiler a bit of reorganization in fonts and whatever, but I wasn't able to gather more than this.

    thanks

    Ricky
     
  13. Sep 21, 2013 #12

    mathwonk

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    usually different editions of books do not matter. earlier is usually better in my opinion, but later ones may have fewer typos. in this case i do not know any but the first editions.
     
  14. Sep 21, 2013 #13
    Thanks,
    I really like this quite old fashioned version, in which figures are all handmade. Also I've read here the most important volumes are the first 2, that's why I had almost no doubt in buying this low priced used offer.
    On a first bird fly on 1st volume, it really impressed me: it does not only talk about diff geom indeed, but it
    treats manifold also from an algebraic topology point of view, but in a sort of handy and intuitive way.
    This sounds good to me, since I'm studying alg top in parallel ( using "The shape of space" and the Hatcher
    bible as well ).

    Thanks

    Ricky


    Regards
     
  15. Sep 25, 2013 #14
    Hi mathwonk,
    I've seen in one of the preface, Spivak is suggesting its Calculus on Manifolds as a prerequisite. I'm addressing this question directly to you since I've seen from other posts of yours you are mentioning about it as one of the best book among your collection.
    I'm not going to go off topic here, I'm just trying to understand if I really need reading that book, since now I own the 2 volumes of this series. I'm mainly interested in manifolds, since I'm studying in parallel GR and I'd like to have a more "formal" approach to differential geometry. I've looked at Calculus on manifolds table of content, and I've seen it depicts manifolds as introduction just at its final chapters, landing to Stokes theorem at last.
    From what I can see from the "comprehensive etc etc" first volume table of content, this is actually overlapping almost entirely with those last chapters of Calculus on manifolds. I know Calculus on manifolds is plenty of good exercises, but I was wondering if it worth to buy something which is already covered from this differential series.

    I know also first chapters of Calculus on manifolds are dedicated to multivariable analysis, but I've already almost a good basis on it, mainly due to the courses I attended when I studied Nuclear Engineering in Turin ( in Italy this is almost covered to Analysis I and II courses, while Analysis III is covering complex f()s analysis and functional spaces ).

    I start to read the 1st volume and it seems to me I can understand it pretty fine, so may be these 2 volumes can be enough to me. I was also evaluating Munkres' Analysis on manifolds, but I saw so many different opinion about. But again, since I'm not interested in a total refresh of Advanced Calculus, these 2 volumes could fit my needs, may be.

    I'd like to highlight, I'm a sort of physics/maths books collector ( even I'm by far from your 350 library ) so this could fit perfectly among "classical" texts. Also, it's not really expensive ( Munkres almost costs twice here in Italy ), so I could face this purchase as well. I was just wondering if it really adds to new content to this differential series.

    Thanks, regards

    R.
     
    Last edited: Sep 25, 2013
  16. Sep 28, 2013 #15

    mathwonk

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    mike's calculus on manifolds is a 140 page treatment of several variable calculus, specifically, derivatives and the implicit function theorem; riemann integration and fubini's theorem; differential forms and exterior derivatives; stokes theorem on manifolds (using partitions of unity to glue together copies of the theorem on rectangles

    if you already know advanced calculus, and you seem to, i would just keep on reading diff geom and refer back to calc on manifolds only if a gap in background comes up.

    comparing the two books just now reminds me that diff geom seems only to use the first three chapters of calc on manifolds, i.e. the basic theory of diff and integ.

    the chapters 4 and 5 of calc on manifolds (diff forms and integration on manifolds, poincare's lemma, stokes' theorem) seems to be covered again by chapters 4 and 8 of diff geom vol 1.
     
  17. Oct 4, 2013 #16
    Thanks mathwonk, this was my idea as well. I've noticed the part on manifolds was actually overlapping between them, that's why I posted my initial question.
    I will postpone Calculus on Manifolds as fas as it will become really needed, eventually. I'm allocating some money for a bunch of books every month, and since I'm actually studying in parallel the Lorentz group representations, especially looking at its connection with 2 spinors and mobius transformations, I've just bought "Representation theory: a first course", by Fulton ( may be you have seen my post about twistor theory. I'm really excited in reading Penrose book about them, and I really think its theory about spinor in relativity, inhereted by Cartan's first idea, has been wrongly a bit abandoned, without really getting all its powerful content due to complex num structures as companion. But first judging the history, I have to improve my knowledge about that, of course ).
    I really lack in a strong base about group representation and Lie group/algebra. That's why I prefer that to Calculus on Manifold, hoping I could use the main diff geom ones would be enough to me.

    thanks
     
    Last edited: Oct 4, 2013
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