A Comprehensive Introduction to Differential Geometry, Volume Five, Third Edition by Michael Spivak is the culmination of the series — containing the Gauss-Bonnet-Chern theorem, the comprehensive bibliography of the differential geometry literature, and the final four chapters of Spivak's unified work — available here direct from Publish or Perish, Inc., the official publisher.
Volume Five completes the arc that began in Volume One. It contains Chapters 10 through 13 of the unified work spanning Volumes Three through Five. Partial differential equations, which have threaded through the entire series since their first appearance in Chapter 6 of Volume One, are finally given their full treatment in Chapter 10. The series closes with Chapter 13 and the Gauss-Bonnet-Chern theorem — the place of honor Spivak reserved for it at the end of the book.
What This Volume Covers
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Chapter 10 — And Now a Brief Message from Our Sponsor (PDEs): A self-contained treatment of partial differential equations reaching precisely the results needed for the next two chapters. Covers linear, quasi-linear, and general first order PDEs — characteristic curves, Monge cones, the Cauchy problem, and characteristic initial data. Free initial manifolds for higher order equations. Systems of first order PDEs. The Cauchy-Kowalewski theorem. Classification of second order PDEs — semi-linear and general — with reduction to normal forms. The prototypical PDEs of physics: the wave equation, heat equation, and Laplace's equation. Hyperbolic systems in two variables. Elliptic solutions of second order equations in two variables. Addenda on the Cartan-Kahler theorem and an elementary maximum principle.
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Chapter 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. The Burstin-Janet-Cartan theorem. An addendum on the embedding problem via differential systems.
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Chapter 12 — Rigidity: Rigidity in higher dimensions and type number. Bendings, warpings, and infinitesimal bendings. Minkowski's formulas. Infinitesimal rigidity of convex surfaces. The theorems of Cohn-Vossen, Minkowski, and Christoffel. Local rigidity problems, the role of asymptotic curves, and other classical results. E. E. Levi's theorems and Schilt's theorem. Surfaces in spheres and hyperbolic space. Rigidity for higher codimension.
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Chapter 13 — The Generalized Gauss-Bonnet Theorem and What It Means for Mankind: Historical remarks. Operations on bundles — Whitney sums, induced bundles, the covering homotopy theorem. Grassmannians and universal bundles. The Pfaffian. The Euler class and the Gauss-Bonnet-Chern theorem. Characteristic classes. The cohomology of homogeneous spaces and oriented Grassmannians. Classical invariant theory and the Capelli identities. Pontryagin classes. The Weil homomorphism. Complex bundles, Hermitian inner products, the unitary group, complex Grassmannians, and Chern classes. Relations between Chern, Pontryagin, and Euler classes. A valedictory.
The Bibliography. Volume Five contains the comprehensive bibliography promised throughout the series — organized into other topics in differential geometry, books, and journal articles. It gives some indication, as Spivak noted, of how much has necessarily been left out of even a five-volume work on the subject.
Volume Five requires the foundations of all four preceding volumes. The complete five-volume set is also available at a 20% discount.
About This Edition
Each volume is hardcover with a matte laminate finish. The pages are printed on a premium 60lb uncoated text stock selected for its opacity, smoothness, and reading comfort. At 96 brightness with enhanced opacity, it renders mathematical notation and fine print with sharp contrast and minimal show-through. It is grain-short, meaning the paper fibers run parallel to the spine, which is why the pages turn easily and the book lies flat without being forced. Being acid-free, it will not yellow with age. Each volume is bound using PUR (Polyurethane Reactive) adhesive — a high-performance binding method that offers significantly greater page-pull strength and superior lay-flat quality compared to traditional glues, and holds reliably under heavy use.
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About the Author
Michael Spivak (1940–2020) earned his Ph.D. from Princeton University and is celebrated for writing mathematics textbooks of extraordinary rigor and clarity. A Comprehensive Introduction to Differential Geometry is his magnum opus — a five-volume work that remains the definitive treatment of the subject. He founded Publish or Perish, Inc., through which all of his major works are published.
