A Comprehensive Introduction to Differential Geometry, Volume One, Third Edition by Michael Spivak is the foundational volume of the definitive series on the subject — available here direct from Publish or Perish, Inc., the official publisher.
Volume One establishes the complete modern language of differential geometry, devoting itself entirely to the theory of differentiable manifolds. Spivak's design was deliberate: rather than forcing classical geometric ideas into elementary terms through unnecessary contortions, this volume gives the reader the modern framework — manifolds, bundles, forms, vector fields — that was developed precisely to make those ideas rigorous. With that foundation in hand, the classical works of Gauss, Riemann, and their successors become readable on their own terms.
About This Volume
Spivak wrote in the original preface that any serious introduction to differential geometry faces a fundamental dilemma. Modern treatments are clean and precise but leave the reader unable to read classical works and ignorant of how anyone ever arrived at the definitions. Classical treatments are historically rich but inaccessible to a reader trained in modern mathematics. His solution was to build the modern language first, compare it carefully with classical language throughout, and then — in the volumes that follow — use that foundation to read the foundational papers of Gauss and Riemann directly.
The result is a volume that is both a rigorous introduction to differentiable manifolds and a preparation for the historical and geometric material that follows. The third edition has been fully typeset with redrawn figures, and includes corrections brought to Spivak's attention by readers over many years. He described it as the third and final edition.
What This Volume Covers
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Chapter 1 — Manifolds: Elementary properties and examples of manifolds.
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Chapter 2 — Differential Structures: Smooth structures, smooth functions, partial derivatives, critical points, immersion theorems, and partitions of unity.
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Chapter 3 — The Tangent Bundle: Tangent spaces, vector bundles, vector fields, orientation, and equivalence classes of curves and derivations.
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Chapter 4 — Tensors: The dual bundle, differentials of functions, multilinear functions, covariant and contravariant tensors, mixed tensors, and contraction.
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Chapter 5 — Vector Fields and Differential Equations: Integral curves, existence and uniqueness theorems, local flows, one-parameter groups of diffeomorphisms, Lie derivatives, and brackets.
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Chapter 6 — Integral Manifolds: Classical integrability theorems and the Frobenius integrability theorem, local and global theory.
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Chapter 7 — Differential Forms: Alternating functions, the wedge product, forms, closed and exact forms, and the Poincare Lemma.
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Chapter 8 — Integration: Classical line and surface integrals, Stokes' theorem, integrals over manifolds, volume elements, and de Rham cohomology.
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Chapter 9 — Riemannian Metrics: Inner products, Riemannian metrics, length of curves, the calculus of variations, geodesics, the exponential map, and geodesic completeness.
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Chapter 10 — Lie Groups: Lie groups, Lie algebras, one-parameter subgroups, the exponential map, closed subgroups, left invariant forms, bi-invariant metrics, and the equations of structure.
Volume One is the essential starting point for the series. Volumes Two through Five build directly on the foundation established here, moving into the classical geometry of Gauss and Riemann, curvature theory, higher-dimensional geometry, and partial differential equations. 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.
