A Comprehensive Introduction to Differential Geometry, Volume Three, Third Edition by Michael Spivak is essentially a complete course in classical surface theory — available here direct from Publish or Perish, Inc., the official publisher.
Spivak described Volumes Three, Four, and Five as constituting a single volume, with Volume Three covering Chapters 1 through 6. Where the first two volumes built the language and encountered the foundational papers of Gauss and Riemann, Volume Three puts that preparation to work — moving through the full theory of surfaces in space, the geometry of submanifolds, and the role that partial differential equations have played throughout the series since they were first introduced in Chapter 6 of Volume One.
About This Volume
Volume Three is, as Spivak noted in the preface, essentially a course in classical surface theory. Chapter 1 prepares the ground by applying the intrinsic geometry of Riemannian manifolds developed in Volume Two to the setting of surfaces. The remaining chapters work through the full classical theory, with selected modern results — including Kuiper's theorem and work of Hartman and Nirenberg, Massey, and Maltz — woven in where the classical development leads naturally to them.
Partial differential equations have played a decisive role throughout the series, and Volume Three is no exception. At times the equations are suppressed in favor of a more geometric conception involving distributions; at other times they are expressed in terms of differential forms. Both approaches are present and compared. Problems are restricted to the absolute minimum — essentially facts left to the reader as exercises.
What This Volume Covers
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Chapter 1 — Intrinsic Geometry of Riemannian Manifolds: Prepares the ground for surface theory by applying the Riemannian geometry of Volume Two — geodesics, curvature, and the exponential map — to the submanifold setting.
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Chapter 2 — Surfaces in Space: The classical theory of surfaces — principal curvatures, lines of curvature, the shape operator, umbilics, surfaces of revolution, minimal surfaces, and ruled surfaces. The second half, from page 75, may be omitted without loss of continuity.
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Chapter 3 — The Fundamental Theorem of Surface Theory: Existence and uniqueness results for surfaces with prescribed first and second fundamental forms, with applications and generalizations.
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Chapter 4 — Geodesics on Surfaces: Geodesics, the Gauss-Bonnet theorem for surfaces, triangulations, the Euler characteristic, and global results including the Hopf Umlaufsatz.
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Chapter 5 — Global Surface Theory: Complete surfaces of constant curvature, compact surfaces of constant negative curvature, the uniformization theorem, and results of Kuiper, Hartman-Nirenberg, Massey, and Maltz.
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Chapter 6 — The Frobenius Theorem Revisited: A summary of how the Frobenius theorem has threaded through the entire series — in Lie groups, affine curve and surface theory, the Test Case proofs, the Fundamental Theorem of Surface Theory, and beyond.
Volume Three requires the foundations of Volumes One and Two. It is the first of three volumes — Three, Four, and Five — that Spivak regarded as constituting a single unified work. 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.
