Spivak, A Comprehensive Introduction to Differential Geometry
Volume 5.   3rd edition. x + 467 pages.  Clothbound.  1999

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Outline of Contents:

10. And Now A Brief Message From Our Sponsor 
        1. FIRST ORDER PDE's
            Linear first order PDE's; characteristic curves; Cauchy problem for
            free initial curves. Quasi-linear first order PDE's; characteristic
            curves; Cauchy problem for 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.
        2. FREE INITIAL MANIFOLDS FOR HIGHER ORDER EQUATIONS
        3. SYSTEMS OF FIRST ORDER PDE'S
        4. THE CAUCHY-KOWALEWSKI THEOREM 
        5. CLASSIFICATION OF SECOND ORDER PDE'S
            Classification of semi-linear equations. Reduction to normal forms.
            Classification of general second order equations.
        6. THE PROTOTYPICAL PDE'S OF PHYSICS
            The wave equation; the heat equation; Laplace's equation. Elementary
            properties.
    7. HYPERBOLIC SYSTEMS IN TWO VARIABLES
    8. HYPERBOLIC SECOND ORDER EQUATIONS IN TWO VARIABLES
            First reduction of the problem. New system of characteristic equations.
            Characteristic initial data. Monge-Ampère equations.
    9. ELLIPTIC SOLUTIONS OF SECOND ORDER EQUATIONS IN TWO VARIABLES
   Addenda. Differential systems; the Cartan-Kähler Theorem. An elementary
    maximum principal.
11. Existence and Non-Existence of Isometric Imbeddings
        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.

12. Rigidity
        Rigidity in higher dimensions; type number. Bendings, warpings, and
        infinitesimal bendings. Vector-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 the 3-sphere and hyperbolic
        3 space. Rigidity for higher codimension.
        Addendum. Infinitesimal bendings of rotation surfaces.

13. The Generalized Gauss-Bonnet Theorem 
        Historical remarks.
         1. OPERATIONS ON BUNDLES
            Bundle maps and principal bundle maps; Whitney sums and induced
            bundles; the covering homotopy theorem.
         2. GRASSMANNIANS AND UNIVERSAL BUNDLES
         3. THE PFAFFIAN
         4. DEFINING THE EULER CLASS IN TERMS OF A CONNECTION
            The Euler class. The class C. The Gauss-Bonnet-Chern Theorem.
         5. THE CONCEPT OF CHARACTERISTIC CLASSES
         6. THE COHOMOLOGY OF HOMOGENEOUS SPACES
            The smooth structure of homogeneous spaces. Invariant forms.
         7. A SMATTERING OF CLASSICAL INVARIANT THEORY
            The Capelli identities. The first fundamental theorem of invariant theory
            for O(n) and SO(n).

         8. AN EASIER INVARIANCE PROBLEM 
         9. THE COHOMOLOGY OF THE ORIENTED GRASSMANNIANS
            Computation of the cohomology; Pontryagin classes. Describing the
            characteristic classes in terms of a connection.
        10. THE WEIL HOMOMORPHISM
        11. COMPLEX BUNDLES
            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.
        12. VALEDICTORY 
        Addenda. Invariant theory for the unitary group
        Recovering the differential forms; the Gauss-Bonnet-Chern Theorem
        for manifolds-with-boundary

 BIBLIOGRAPHY
       A. Other topics in Differential Geometry
        B. Books
        C. Journal articles

From reviews:

Stephanie Alexander, Bulletin of the AMS, volume 84, number 1, January 1978

For the 5 volume set:

The Comprehensive introduction is probably best suited for leisurely and enjoyable background reference by almost anyone interested in differential geometry. Great care has been taken to make it accessible to beginners, but even the most seasoned reader will find stimulating reading here ... The appeal of the book is due first of all to its choice of material, which is guided by the liveliest geometric curiosity. In addition, Spivak has a clear, natural and well-motivated style of exposition; in many places, his book unfolds like a novel.

The Comprehensive introduction will be widely read and enjoyed, and will surely become a standard reference for graduate courses in differential geometry. Spivak is greatly to be thanked for this spontaneous, exuberant and beautifully geometrical book.

Specifically for Volume 5:

Spivak has prepared a course on PDE for geometers ... its chosen topics are treated rigorously and in considerable generality. As always, Spivak emphasizes conceptually appealing proofs. 

Several of the book's outstanding virtues are represented in this treatment [of the relationship between characteristic classes and curvature]: it is self-contained; it gives more than cursory attention to classical invariant theory; and it prizes and imparts geometric insight.
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