{"title":"A Comprehensive Introduction to Differential Geometry","description":null,"products":[{"product_id":"a-comprehensive-introduction-to-differential-geometry-vol-1-3rd-edition","title":"A Comprehensive Introduction to Differential Geometry, Vol. 1, 3rd Edition by Michael Spivak | Publish or Perish","description":"\u003c!-- ============================================================\n     A COMPREHENSIVE INTRODUCTION TO DIFFERENTIAL GEOMETRY\n     VOLUME ONE, THIRD EDITION\n     PRODUCT DESCRIPTION — Publish or Perish, Inc.\n     Paste into Shopify product description (HTML editor)\n     No emojis | No copyright symbols in body | Feed-safe\n\n     PAGE TITLE (SEO):\n     A Comprehensive Introduction to Differential Geometry, Vol. 1 by Michael Spivak | Publish or Perish\n\n     META DESCRIPTION (156 characters — paste into SEO field):\n     A Comprehensive Introduction to Differential Geometry Vol. 1 by Michael Spivak — the definitive foundation text, direct from the publisher. Free FedEx 2Day on US orders.\n     ============================================================ --\u003e\n\u003cstyle\u003e\n  .pop-desc {\n    font-family: Georgia, 'Times New Roman', serif;\n    color: #1a1a1a;\n    line-height: 1.75;\n    max-width: 720px;\n  }\n  .pop-desc p {\n    margin: 0 0 1.25em 0;\n    font-size: 1em;\n  }\n  .pop-desc strong {\n    font-weight: 700;\n  }\n  .pop-desc em {\n    font-style: italic;\n  }\n  \/* Opening lead paragraph *\/\n  .pop-desc .pop-lead {\n    font-size: 1.05em;\n    border-left: 3px solid #1a1a1a;\n    padding-left: 1.1em;\n    margin-bottom: 1.75em;\n  }\n  \/* Section headings *\/\n  .pop-desc h3 {\n    font-family: Georgia, 'Times New Roman', serif;\n    font-size: 0.8em;\n    font-weight: 700;\n    letter-spacing: 0.12em;\n    text-transform: uppercase;\n    color: #1a1a1a;\n    margin: 2em 0 0.75em 0;\n    padding-bottom: 0.4em;\n    border-bottom: 1px solid #d0d0d0;\n  }\n  \/* Feature list *\/\n  .pop-desc ul.pop-features {\n    list-style: none;\n    padding: 0;\n    margin: 0 0 1.5em 0;\n  }\n  .pop-desc ul.pop-features li {\n    padding: 0.55em 0 0.55em 1.4em;\n    position: relative;\n    border-bottom: 1px solid #efefef;\n    font-size: 0.97em;\n  }\n  .pop-desc ul.pop-features li:last-child {\n    border-bottom: none;\n  }\n  .pop-desc ul.pop-features li::before {\n    content: '';\n    position: absolute;\n    left: 0;\n    top: 50%;\n    transform: translateY(-50%);\n    width: 5px;\n    height: 5px;\n    background: #1a1a1a;\n    border-radius: 50%;\n  }\n  \/* Chapter list *\/\n  .pop-desc .pop-chapters {\n    list-style: none;\n    padding: 0;\n    margin: 0 0 1.5em 0;\n  }\n  .pop-desc .pop-chapters li {\n    padding: 0.6em 0 0.6em 1.4em;\n    position: relative;\n    border-bottom: 1px solid #efefef;\n    font-size: 0.95em;\n  }\n  .pop-desc .pop-chapters li:last-child {\n    border-bottom: none;\n  }\n  .pop-desc .pop-chapters li::before {\n    content: '';\n    position: absolute;\n    left: 0;\n    top: 50%;\n    transform: translateY(-50%);\n    width: 5px;\n    height: 5px;\n    background: #1a1a1a;\n    border-radius: 50%;\n  }\n  \/* Series note box *\/\n  .pop-desc .pop-note {\n    background: #f7f7f5;\n    border-left: 3px solid #1a1a1a;\n    padding: 1em 1.25em;\n    margin: 1.75em 0;\n    font-size: 0.93em;\n  }\n  .pop-desc .pop-note p {\n    margin: 0;\n  }\n  \/* Shipping strip *\/\n  .pop-desc .pop-shipping {\n    display: flex;\n    flex-wrap: wrap;\n    gap: 0;\n    border: 1px solid #d0d0d0;\n    margin: 1.75em 0;\n    font-size: 0.8em;\n  }\n  .pop-desc .pop-shipping-item {\n    flex: 1 1 auto;\n    min-width: 0;\n    padding: 0.7em 0.5em;\n    text-align: center;\n    border-right: 1px solid #d0d0d0;\n    box-sizing: border-box;\n  }\n  .pop-desc .pop-shipping-item:last-child {\n    border-right: none;\n  }\n  .pop-desc .pop-shipping-item .pop-shipping-label {\n    display: block;\n    font-size: 0.75em;\n    letter-spacing: 0.08em;\n    text-transform: uppercase;\n    color: #777;\n    margin-bottom: 0.2em;\n    white-space: nowrap;\n  }\n  .pop-desc .pop-shipping-item .pop-shipping-value {\n    display: block;\n    font-weight: 700;\n    color: #1a1a1a;\n    word-break: break-word;\n  }\n  \/* Divider *\/\n  .pop-desc hr.pop-rule {\n    border: none;\n    border-top: 1px solid #d0d0d0;\n    margin: 2em 0;\n  }\n  \/* Author bio *\/\n  .pop-desc .pop-author {\n    font-size: 0.93em;\n    color: #333;\n  }\n\u003c\/style\u003e\n\u003cdiv itemtype=\"https:\/\/schema.org\/Book\" itemscope=\"\" class=\"pop-desc\"\u003e\n\u003c!-- Plain SEO opener — keyword leads --\u003e\n\u003cp\u003e\u003cstrong itemprop=\"name\"\u003eA Comprehensive Introduction to Differential Geometry, Volume One, Third Edition by Michael Spivak\u003c\/strong\u003e is the foundational volume of the definitive series on the subject — available here direct from \u003cspan itemprop=\"publisher\"\u003ePublish or Perish, Inc.\u003c\/span\u003e, the official publisher.\u003c\/p\u003e\n\u003cp class=\"pop-lead\"\u003eVolume 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.\u003c\/p\u003e\n\u003chr class=\"pop-rule\"\u003e\n\u003ch3\u003eAbout This Volume\u003c\/h3\u003e\n\u003cp\u003eSpivak 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.\u003c\/p\u003e\n\u003cp\u003eThe 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.\u003c\/p\u003e\n\u003chr class=\"pop-rule\"\u003e\n\u003ch3\u003eWhat This Volume Covers\u003c\/h3\u003e\n\u003cul class=\"pop-chapters\"\u003e\n\u003cli\u003e\n\u003cstrong\u003eChapter 1 — Manifolds:\u003c\/strong\u003e Elementary properties and examples of manifolds.\u003c\/li\u003e\n\u003cli\u003e\n\u003cstrong\u003eChapter 2 — Differential Structures:\u003c\/strong\u003e Smooth structures, smooth functions, partial derivatives, critical points, immersion theorems, and partitions of unity.\u003c\/li\u003e\n\u003cli\u003e\n\u003cstrong\u003eChapter 3 — The Tangent Bundle:\u003c\/strong\u003e Tangent spaces, vector bundles, vector fields, orientation, and equivalence classes of curves and derivations.\u003c\/li\u003e\n\u003cli\u003e\n\u003cstrong\u003eChapter 4 — Tensors:\u003c\/strong\u003e The dual bundle, differentials of functions, multilinear functions, covariant and contravariant tensors, mixed tensors, and contraction.\u003c\/li\u003e\n\u003cli\u003e\n\u003cstrong\u003eChapter 5 — Vector Fields and Differential Equations:\u003c\/strong\u003e Integral curves, existence and uniqueness theorems, local flows, one-parameter groups of diffeomorphisms, Lie derivatives, and brackets.\u003c\/li\u003e\n\u003cli\u003e\n\u003cstrong\u003eChapter 6 — Integral Manifolds:\u003c\/strong\u003e Classical integrability theorems and the Frobenius integrability theorem, local and global theory.\u003c\/li\u003e\n\u003cli\u003e\n\u003cstrong\u003eChapter 7 — Differential Forms:\u003c\/strong\u003e Alternating functions, the wedge product, forms, closed and exact forms, and the Poincare Lemma.\u003c\/li\u003e\n\u003cli\u003e\n\u003cstrong\u003eChapter 8 — Integration:\u003c\/strong\u003e Classical line and surface integrals, Stokes' theorem, integrals over manifolds, volume elements, and de Rham cohomology.\u003c\/li\u003e\n\u003cli\u003e\n\u003cstrong\u003eChapter 9 — Riemannian Metrics:\u003c\/strong\u003e Inner products, Riemannian metrics, length of curves, the calculus of variations, geodesics, the exponential map, and geodesic completeness.\u003c\/li\u003e\n\u003cli\u003e\n\u003cstrong\u003eChapter 10 — Lie Groups:\u003c\/strong\u003e Lie groups, Lie algebras, one-parameter subgroups, the exponential map, closed subgroups, left invariant forms, bi-invariant metrics, and the equations of structure.\u003c\/li\u003e\n\u003c\/ul\u003e\n\u003chr class=\"pop-rule\"\u003e\n\u003cdiv class=\"pop-note\"\u003e\n\u003cp\u003eVolume 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.\u003c\/p\u003e\n\u003c\/div\u003e\n\u003chr class=\"pop-rule\"\u003e\n\u003ch3\u003eAbout This Edition\u003c\/h3\u003e\n\u003cp\u003eEach 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.\u003c\/p\u003e\n\u003chr class=\"pop-rule\"\u003e\n\u003c!-- Shipping strip --\u003e\n\u003cdiv class=\"pop-shipping\"\u003e\n\u003cdiv class=\"pop-shipping-item\"\u003e\n\u003cspan class=\"pop-shipping-label\"\u003eShipping\u003c\/span\u003e \u003cspan class=\"pop-shipping-value\"\u003eFree FedEx 2Day (US)\u003c\/span\u003e\n\u003c\/div\u003e\n\u003cdiv class=\"pop-shipping-item\"\u003e\n\u003cspan class=\"pop-shipping-label\"\u003eProcessing\u003c\/span\u003e \u003cspan class=\"pop-shipping-value\"\u003e1 Business Day\u003c\/span\u003e\n\u003c\/div\u003e\n\u003cdiv class=\"pop-shipping-item\"\u003e\n\u003cspan class=\"pop-shipping-label\"\u003eDelivery\u003c\/span\u003e \u003cspan class=\"pop-shipping-value\"\u003e2 to 4 Days\u003c\/span\u003e\n\u003c\/div\u003e\n\u003cdiv class=\"pop-shipping-item\"\u003e\n\u003cspan class=\"pop-shipping-label\"\u003eReturns\u003c\/span\u003e \u003cspan class=\"pop-shipping-value\"\u003e30 Days\u003c\/span\u003e\n\u003c\/div\u003e\n\u003c\/div\u003e\n\u003chr class=\"pop-rule\"\u003e\n\u003ch3\u003eAbout the Author\u003c\/h3\u003e\n\u003cp itemprop=\"about\" class=\"pop-author\"\u003eMichael Spivak (1940–2020) earned his Ph.D. from Princeton University and is celebrated for writing mathematics textbooks of extraordinary rigor and clarity. \u003cem\u003eA Comprehensive Introduction to Differential Geometry\u003c\/em\u003e 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.\u003c\/p\u003e\n\u003c\/div\u003e","brand":"Publish or Perish, Inc. ®","offers":[{"title":"Default Title","offer_id":50005562556718,"sku":"POP-ACIDG-V1-3E","price":112.99,"currency_code":"USD","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0829\/3705\/3486\/files\/DG1covercrop.png?v=1763683769"},{"product_id":"a-comprehensive-introduction-to-differential-geometry-vol-2-3rd-edition","title":"A Comprehensive Introduction to Differential Geometry, Vol. 2, 3rd Edition by Michael Spivak | Publish or Perish","description":"\u003c!-- ============================================================\n     A COMPREHENSIVE INTRODUCTION TO DIFFERENTIAL GEOMETRY\n     VOLUME TWO, THIRD EDITION\n     PRODUCT DESCRIPTION — Publish or Perish, Inc.\n     Paste into Shopify product description (HTML editor)\n     No emojis | No copyright symbols in body | Feed-safe\n\n     PAGE TITLE (SEO):\n     A Comprehensive Introduction to Differential Geometry, Vol. 2, 3rd Edition by Michael Spivak | Publish or Perish\n\n     META DESCRIPTION (156 characters — paste into SEO field):\n     A Comprehensive Introduction to Differential Geometry Vol. 2 by Michael Spivak — curves, surfaces, Gauss, and Riemann. Direct from the publisher. Free FedEx 2Day on US orders.\n     ============================================================ --\u003e\n\n\u003cstyle\u003e\n  .pop-desc {\n    font-family: Georgia, 'Times New Roman', serif;\n    color: #1a1a1a;\n    line-height: 1.75;\n    max-width: 720px;\n  }\n  .pop-desc p {\n    margin: 0 0 1.25em 0;\n    font-size: 1em;\n  }\n  .pop-desc strong {\n    font-weight: 700;\n  }\n  .pop-desc em {\n    font-style: italic;\n  }\n  \/* Opening lead paragraph *\/\n  .pop-desc .pop-lead {\n    font-size: 1.05em;\n    border-left: 3px solid #1a1a1a;\n    padding-left: 1.1em;\n    margin-bottom: 1.75em;\n  }\n  \/* Section headings *\/\n  .pop-desc h3 {\n    font-family: Georgia, 'Times New Roman', serif;\n    font-size: 0.8em;\n    font-weight: 700;\n    letter-spacing: 0.12em;\n    text-transform: uppercase;\n    color: #1a1a1a;\n    margin: 2em 0 0.75em 0;\n    padding-bottom: 0.4em;\n    border-bottom: 1px solid #d0d0d0;\n  }\n  \/* Chapter list *\/\n  .pop-desc .pop-chapters {\n    list-style: none;\n    padding: 0;\n    margin: 0 0 1.5em 0;\n  }\n  .pop-desc .pop-chapters li {\n    padding: 0.6em 0 0.6em 1.4em;\n    position: relative;\n    border-bottom: 1px solid #efefef;\n    font-size: 0.95em;\n  }\n  .pop-desc .pop-chapters li:last-child {\n    border-bottom: none;\n  }\n  .pop-desc .pop-chapters li::before {\n    content: '';\n    position: absolute;\n    left: 0;\n    top: 50%;\n    transform: translateY(-50%);\n    width: 5px;\n    height: 5px;\n    background: #1a1a1a;\n    border-radius: 50%;\n  }\n  \/* Series note box *\/\n  .pop-desc .pop-note {\n    background: #f7f7f5;\n    border-left: 3px solid #1a1a1a;\n    padding: 1em 1.25em;\n    margin: 1.75em 0;\n    font-size: 0.93em;\n  }\n  .pop-desc .pop-note p {\n    margin: 0;\n  }\n  \/* Shipping strip *\/\n  .pop-desc .pop-shipping {\n    display: flex;\n    flex-wrap: wrap;\n    gap: 0;\n    border: 1px solid #d0d0d0;\n    margin: 1.75em 0;\n    font-size: 0.8em;\n  }\n  .pop-desc .pop-shipping-item {\n    flex: 1 1 auto;\n    min-width: 0;\n    padding: 0.7em 0.5em;\n    text-align: center;\n    border-right: 1px solid #d0d0d0;\n    box-sizing: border-box;\n  }\n  .pop-desc .pop-shipping-item:last-child {\n    border-right: none;\n  }\n  .pop-desc .pop-shipping-item .pop-shipping-label {\n    display: block;\n    font-size: 0.75em;\n    letter-spacing: 0.08em;\n    text-transform: uppercase;\n    color: #777;\n    margin-bottom: 0.2em;\n    white-space: nowrap;\n  }\n  .pop-desc .pop-shipping-item .pop-shipping-value {\n    display: block;\n    font-weight: 700;\n    color: #1a1a1a;\n    word-break: break-word;\n  }\n  \/* Divider *\/\n  .pop-desc hr.pop-rule {\n    border: none;\n    border-top: 1px solid #d0d0d0;\n    margin: 2em 0;\n  }\n  \/* Author bio *\/\n  .pop-desc .pop-author {\n    font-size: 0.93em;\n    color: #333;\n  }\n\u003c\/style\u003e\n\n\u003cdiv class=\"pop-desc\" itemscope itemtype=\"https:\/\/schema.org\/Book\"\u003e\n\n  \u003c!-- Plain SEO opener — keyword leads --\u003e\n  \u003cp\u003e\u003cstrong itemprop=\"name\"\u003eA Comprehensive Introduction to Differential Geometry, Volume Two, Third Edition by Michael Spivak\u003c\/strong\u003e is where the geometry begins in earnest — available here direct from \u003cspan itemprop=\"publisher\"\u003ePublish or Perish, Inc.\u003c\/span\u003e, the official publisher.\u003c\/p\u003e\n\n  \u003cp class=\"pop-lead\"\u003e\n    If Volume One established the modern language of differentiable manifolds, Volume Two puts that language to work. Beginning with the simplest geometric objects — curves in the plane and in space — it follows the semi-historical path promised in the first volume, moving through the classical surface theory of Euler and Gauss, the foundational papers of Riemann, and the development of curvature theory in its modern form. The most decisive encounters with classical differential geometry are here, in Chapters 3 and 4, where Gauss and Riemann are read on their own terms.\n  \u003c\/p\u003e\n\n  \u003chr class=\"pop-rule\"\u003e\n\n  \u003ch3\u003eAbout This Volume\u003c\/h3\u003e\n\n  \u003cp\u003e\n    Spivak noted in the preface that this volume begins the study of modern differential geometry in earnest. It follows the historical path laid out in Volume One — not as a history lesson, but as the most illuminating way to understand where the concepts came from and why they take the forms they do. Chapter 3 presents Gauss' theory of surfaces directly, including a guide to reading Gauss himself. Chapter 4 does the same for Riemann, including a treatment of his inaugural lecture and the birth of the Riemann curvature tensor. Spivak's view was that skipping these classical encounters misses all the fun.\n  \u003c\/p\u003e\n\n  \u003cp\u003e\n    There are no problem sets in this volume — the material does not easily lend itself to them. The final volume of the series contains a comprehensive bibliography of the differential geometry literature, including texts where problems may be found.\n  \u003c\/p\u003e\n\n  \u003chr class=\"pop-rule\"\u003e\n\n  \u003ch3\u003eWhat This Volume Covers\u003c\/h3\u003e\n\n  \u003cul class=\"pop-chapters\"\u003e\n    \u003cli\u003e\n\u003cstrong\u003eChapter 1 — Curves in the Plane and in Space:\u003c\/strong\u003e Curvature of plane curves, convex curves, curvature and torsion of space curves, the Serret-Frenet formulas, the natural form on a Lie group, and classification of curves under affine motions.\u003c\/li\u003e\n    \u003cli\u003e\n\u003cstrong\u003eChapter 2 — What They Knew About Surfaces Before Gauss:\u003c\/strong\u003e Euler's Theorem and Meusnier's Theorem — the state of surface theory before Gauss transformed it.\u003c\/li\u003e\n    \u003cli\u003e\n\u003cstrong\u003eChapter 3 — The Curvature of Surfaces in Space:\u003c\/strong\u003e A guide to reading Gauss, followed by Gauss' full theory of surfaces — the Gauss map, Gaussian curvature, the Weingarten map, the first and second fundamental forms, the Theorema Egregium, geodesics, and the integral of curvature over a geodesic triangle.\u003c\/li\u003e\n    \u003cli\u003e\n\u003cstrong\u003eChapter 4 — The Curvature of Higher Dimensional Manifolds:\u003c\/strong\u003e Riemann's inaugural lecture, Riemannian normal coordinates, a prize essay, and the birth of the Riemann curvature tensor — sectional curvature and the conditions for flatness.\u003c\/li\u003e\n    \u003cli\u003e\n\u003cstrong\u003eChapter 5 — The Absolute Differential Calculus (The Ricci Calculus):\u003c\/strong\u003e Covariant derivatives, Ricci's Lemma, Ricci's identities, the curvature tensor, classical connections, the torsion tensor, geodesics, and Bianchi's identities.\u003c\/li\u003e\n    \u003cli\u003e\n\u003cstrong\u003eChapter 6 — The Nabla Operator:\u003c\/strong\u003e Koszul connections, covariant derivatives, parallel translation, the Levi-Civita connection, the curvature tensor, geodesics, and the first variation formula.\u003c\/li\u003e\n    \u003cli\u003e\n\u003cstrong\u003eChapter 7 — The Moving Frame:\u003c\/strong\u003e Moving frames, the structural equations of Euclidean space and Riemannian manifolds, adapted frames, Cartan connections, manifolds of constant curvature, Schur's theorem, and conformally equivalent manifolds.\u003c\/li\u003e\n    \u003cli\u003e\n\u003cstrong\u003eChapter 8 — Connections in Principal Bundles:\u003c\/strong\u003e Principal bundles, Lie groups acting on manifolds, Cartan connections, Ehresmann connections, parallel translation, the curvature form, structural equations, and Bianchi's identities.\u003c\/li\u003e\n  \u003c\/ul\u003e\n\n  \u003chr class=\"pop-rule\"\u003e\n\n  \u003cdiv class=\"pop-note\"\u003e\n    \u003cp\u003eVolume Two requires the foundation established in Volume One. Together, Volumes One and Two form the prerequisite for \u003cem\u003ePhysics for Mathematicians, Mechanics I\u003c\/em\u003e. The complete five-volume set is also available at a 20% discount.\u003c\/p\u003e\n  \u003c\/div\u003e\n\n  \u003chr class=\"pop-rule\"\u003e\n\n  \u003ch3\u003eAbout This Edition\u003c\/h3\u003e\n\n  \u003cp\u003e\n    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.\n  \u003c\/p\u003e\n\n  \u003chr class=\"pop-rule\"\u003e\n\n  \u003c!-- Shipping strip --\u003e\n  \u003cdiv class=\"pop-shipping\"\u003e\n    \u003cdiv class=\"pop-shipping-item\"\u003e\n      \u003cspan class=\"pop-shipping-label\"\u003eShipping\u003c\/span\u003e\n      \u003cspan class=\"pop-shipping-value\"\u003eFree FedEx 2Day (US)\u003c\/span\u003e\n    \u003c\/div\u003e\n    \u003cdiv class=\"pop-shipping-item\"\u003e\n      \u003cspan class=\"pop-shipping-label\"\u003eProcessing\u003c\/span\u003e\n      \u003cspan class=\"pop-shipping-value\"\u003e1 Business Day\u003c\/span\u003e\n    \u003c\/div\u003e\n    \u003cdiv class=\"pop-shipping-item\"\u003e\n      \u003cspan class=\"pop-shipping-label\"\u003eDelivery\u003c\/span\u003e\n      \u003cspan class=\"pop-shipping-value\"\u003e2 to 4 Days\u003c\/span\u003e\n    \u003c\/div\u003e\n    \u003cdiv class=\"pop-shipping-item\"\u003e\n      \u003cspan class=\"pop-shipping-label\"\u003eReturns\u003c\/span\u003e\n      \u003cspan class=\"pop-shipping-value\"\u003e30 Days\u003c\/span\u003e\n    \u003c\/div\u003e\n  \u003c\/div\u003e\n\n  \u003chr class=\"pop-rule\"\u003e\n\n  \u003ch3\u003eAbout the Author\u003c\/h3\u003e\n\n  \u003cp class=\"pop-author\" itemprop=\"about\"\u003e\n    Michael Spivak (1940–2020) earned his Ph.D. from Princeton University and is celebrated for writing mathematics textbooks of extraordinary rigor and clarity. \u003cem\u003eA Comprehensive Introduction to Differential Geometry\u003c\/em\u003e 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.\n  \u003c\/p\u003e\n\n\u003c\/div\u003e","brand":"Publish or Perish, Inc. ®","offers":[{"title":"Default Title","offer_id":49983438487854,"sku":"POP-ACIDG-V2-3E","price":94.99,"currency_code":"USD","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0829\/3705\/3486\/files\/DG2Covercrop.png?v=1763683771"},{"product_id":"a-comprehensive-introduction-to-differential-geometry-vol-3-3rd-edition","title":"A Comprehensive Introduction to Differential Geometry, Vol. 3, 3rd Edition by Michael Spivak | Publish or Perish","description":"\u003c!-- ============================================================\n     A COMPREHENSIVE INTRODUCTION TO DIFFERENTIAL GEOMETRY\n     VOLUME THREE, THIRD EDITION\n     PRODUCT DESCRIPTION — Publish or Perish, Inc.\n     Paste into Shopify product description (HTML editor)\n     No emojis | No copyright symbols in body | Feed-safe\n\n     PAGE TITLE (SEO):\n     A Comprehensive Introduction to Differential Geometry, Vol. 3, 3rd Edition by Michael Spivak | Publish or Perish\n\n     META DESCRIPTION (157 characters — paste into SEO field):\n     A Comprehensive Introduction to Differential Geometry Vol. 3 by Michael Spivak — classical surface theory in full. Direct from the publisher. Free FedEx 2Day on US orders.\n     ============================================================ --\u003e\n\n\u003cstyle\u003e\n  .pop-desc {\n    font-family: Georgia, 'Times New Roman', serif;\n    color: #1a1a1a;\n    line-height: 1.75;\n    max-width: 720px;\n  }\n  .pop-desc p {\n    margin: 0 0 1.25em 0;\n    font-size: 1em;\n  }\n  .pop-desc strong {\n    font-weight: 700;\n  }\n  .pop-desc em {\n    font-style: italic;\n  }\n  .pop-desc .pop-lead {\n    font-size: 1.05em;\n    border-left: 3px solid #1a1a1a;\n    padding-left: 1.1em;\n    margin-bottom: 1.75em;\n  }\n  .pop-desc h3 {\n    font-family: Georgia, 'Times New Roman', serif;\n    font-size: 0.8em;\n    font-weight: 700;\n    letter-spacing: 0.12em;\n    text-transform: uppercase;\n    color: #1a1a1a;\n    margin: 2em 0 0.75em 0;\n    padding-bottom: 0.4em;\n    border-bottom: 1px solid #d0d0d0;\n  }\n  .pop-desc ul.pop-chapters {\n    list-style: none;\n    padding: 0;\n    margin: 0 0 1.5em 0;\n  }\n  .pop-desc ul.pop-chapters li {\n    padding: 0.6em 0 0.6em 1.4em;\n    position: relative;\n    border-bottom: 1px solid #efefef;\n    font-size: 0.95em;\n  }\n  .pop-desc ul.pop-chapters li:last-child {\n    border-bottom: none;\n  }\n  .pop-desc ul.pop-chapters li::before {\n    content: '';\n    position: absolute;\n    left: 0;\n    top: 50%;\n    transform: translateY(-50%);\n    width: 5px;\n    height: 5px;\n    background: #1a1a1a;\n    border-radius: 50%;\n  }\n  .pop-desc .pop-note {\n    background: #f7f7f5;\n    border-left: 3px solid #1a1a1a;\n    padding: 1em 1.25em;\n    margin: 1.75em 0;\n    font-size: 0.93em;\n  }\n  .pop-desc .pop-note p {\n    margin: 0;\n  }\n  .pop-desc .pop-shipping {\n    display: flex;\n    flex-wrap: wrap;\n    gap: 0;\n    border: 1px solid #d0d0d0;\n    margin: 1.75em 0;\n    font-size: 0.8em;\n  }\n  .pop-desc .pop-shipping-item {\n    flex: 1 1 auto;\n    min-width: 0;\n    padding: 0.7em 0.5em;\n    text-align: center;\n    border-right: 1px solid #d0d0d0;\n    box-sizing: border-box;\n  }\n  .pop-desc .pop-shipping-item:last-child {\n    border-right: none;\n  }\n  .pop-desc .pop-shipping-item .pop-shipping-label {\n    display: block;\n    font-size: 0.75em;\n    letter-spacing: 0.08em;\n    text-transform: uppercase;\n    color: #777;\n    margin-bottom: 0.2em;\n    white-space: nowrap;\n  }\n  .pop-desc .pop-shipping-item .pop-shipping-value {\n    display: block;\n    font-weight: 700;\n    color: #1a1a1a;\n    word-break: break-word;\n  }\n  .pop-desc hr.pop-rule {\n    border: none;\n    border-top: 1px solid #d0d0d0;\n    margin: 2em 0;\n  }\n  .pop-desc .pop-author {\n    font-size: 0.93em;\n    color: #333;\n  }\n\u003c\/style\u003e\n\n\u003cdiv class=\"pop-desc\" itemscope itemtype=\"https:\/\/schema.org\/Book\"\u003e\n\n  \u003c!-- Plain SEO opener — keyword leads --\u003e\n  \u003cp\u003e\u003cstrong itemprop=\"name\"\u003eA Comprehensive Introduction to Differential Geometry, Volume Three, Third Edition by Michael Spivak\u003c\/strong\u003e is essentially a complete course in classical surface theory — available here direct from \u003cspan itemprop=\"publisher\"\u003ePublish or Perish, Inc.\u003c\/span\u003e, the official publisher.\u003c\/p\u003e\n\n  \u003cp class=\"pop-lead\"\u003e\n    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.\n  \u003c\/p\u003e\n\n  \u003chr class=\"pop-rule\"\u003e\n\n  \u003ch3\u003eAbout This Volume\u003c\/h3\u003e\n\n  \u003cp\u003e\n    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.\n  \u003c\/p\u003e\n\n  \u003cp\u003e\n    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.\n  \u003c\/p\u003e\n\n  \u003chr class=\"pop-rule\"\u003e\n\n  \u003ch3\u003eWhat This Volume Covers\u003c\/h3\u003e\n\n  \u003cul class=\"pop-chapters\"\u003e\n    \u003cli\u003e\n\u003cstrong\u003eChapter 1 — Intrinsic Geometry of Riemannian Manifolds:\u003c\/strong\u003e Prepares the ground for surface theory by applying the Riemannian geometry of Volume Two — geodesics, curvature, and the exponential map — to the submanifold setting.\u003c\/li\u003e\n    \u003cli\u003e\n\u003cstrong\u003eChapter 2 — Surfaces in Space:\u003c\/strong\u003e 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.\u003c\/li\u003e\n    \u003cli\u003e\n\u003cstrong\u003eChapter 3 — The Fundamental Theorem of Surface Theory:\u003c\/strong\u003e Existence and uniqueness results for surfaces with prescribed first and second fundamental forms, with applications and generalizations.\u003c\/li\u003e\n    \u003cli\u003e\n\u003cstrong\u003eChapter 4 — Geodesics on Surfaces:\u003c\/strong\u003e Geodesics, the Gauss-Bonnet theorem for surfaces, triangulations, the Euler characteristic, and global results including the Hopf Umlaufsatz.\u003c\/li\u003e\n    \u003cli\u003e\n\u003cstrong\u003eChapter 5 — Global Surface Theory:\u003c\/strong\u003e Complete surfaces of constant curvature, compact surfaces of constant negative curvature, the uniformization theorem, and results of Kuiper, Hartman-Nirenberg, Massey, and Maltz.\u003c\/li\u003e\n    \u003cli\u003e\n\u003cstrong\u003eChapter 6 — The Frobenius Theorem Revisited:\u003c\/strong\u003e 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.\u003c\/li\u003e\n  \u003c\/ul\u003e\n\n  \u003chr class=\"pop-rule\"\u003e\n\n  \u003cdiv class=\"pop-note\"\u003e\n    \u003cp\u003eVolume 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.\u003c\/p\u003e\n  \u003c\/div\u003e\n\n  \u003chr class=\"pop-rule\"\u003e\n\n  \u003ch3\u003eAbout This Edition\u003c\/h3\u003e\n\n  \u003cp\u003e\n    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.\n  \u003c\/p\u003e\n\n  \u003chr class=\"pop-rule\"\u003e\n\n  \u003cdiv class=\"pop-shipping\"\u003e\n    \u003cdiv class=\"pop-shipping-item\"\u003e\n      \u003cspan class=\"pop-shipping-label\"\u003eShipping\u003c\/span\u003e\n      \u003cspan class=\"pop-shipping-value\"\u003eFree FedEx 2Day (US)\u003c\/span\u003e\n    \u003c\/div\u003e\n    \u003cdiv class=\"pop-shipping-item\"\u003e\n      \u003cspan class=\"pop-shipping-label\"\u003eProcessing\u003c\/span\u003e\n      \u003cspan class=\"pop-shipping-value\"\u003e1 Business Day\u003c\/span\u003e\n    \u003c\/div\u003e\n    \u003cdiv class=\"pop-shipping-item\"\u003e\n      \u003cspan class=\"pop-shipping-label\"\u003eDelivery\u003c\/span\u003e\n      \u003cspan class=\"pop-shipping-value\"\u003e2 to 4 Days\u003c\/span\u003e\n    \u003c\/div\u003e\n    \u003cdiv class=\"pop-shipping-item\"\u003e\n      \u003cspan class=\"pop-shipping-label\"\u003eReturns\u003c\/span\u003e\n      \u003cspan class=\"pop-shipping-value\"\u003e30 Days\u003c\/span\u003e\n    \u003c\/div\u003e\n  \u003c\/div\u003e\n\n  \u003chr class=\"pop-rule\"\u003e\n\n  \u003ch3\u003eAbout the Author\u003c\/h3\u003e\n\n  \u003cp class=\"pop-author\" itemprop=\"about\"\u003e\n    Michael Spivak (1940–2020) earned his Ph.D. from Princeton University and is celebrated for writing mathematics textbooks of extraordinary rigor and clarity. \u003cem\u003eA Comprehensive Introduction to Differential Geometry\u003c\/em\u003e 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.\n  \u003c\/p\u003e\n\n\u003c\/div\u003e","brand":"Publish or Perish, Inc. ®","offers":[{"title":"Default Title","offer_id":49983418597678,"sku":"POP-ACIDG-V3-3E","price":94.99,"currency_code":"USD","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0829\/3705\/3486\/files\/DG3covercrop.png?v=1763683772"},{"product_id":"a-comprehensive-introduction-to-differential-geometry-vol-4-3rd-edition","title":"A Comprehensive Introduction to Differential Geometry, Vol. 4, 3rd Edition by Michael Spivak | Publish or Perish","description":"\u003c!-- ============================================================\n     A COMPREHENSIVE INTRODUCTION TO DIFFERENTIAL GEOMETRY\n     VOLUME FOUR, THIRD EDITION\n     PRODUCT DESCRIPTION — Publish or Perish, Inc.\n     Paste into Shopify product description (HTML editor)\n     No emojis | No copyright symbols in body | Feed-safe\n\n     PAGE TITLE (SEO):\n     A Comprehensive Introduction to Differential Geometry, Vol. 4, 3rd Edition by Michael Spivak | Publish or Perish\n\n     META DESCRIPTION (158 characters — paste into SEO field):\n     A Comprehensive Introduction to Differential Geometry Vol. 4 by Michael Spivak — higher dimensions, geodesics, minimal surfaces. Direct from the publisher. Free FedEx 2Day on US orders.\n     ============================================================ --\u003e\n\u003cstyle\u003e\n  .pop-desc {\n    font-family: Georgia, 'Times New Roman', serif;\n    color: #1a1a1a;\n    line-height: 1.75;\n    max-width: 720px;\n  }\n  .pop-desc p {\n    margin: 0 0 1.25em 0;\n    font-size: 1em;\n  }\n  .pop-desc strong {\n    font-weight: 700;\n  }\n  .pop-desc em {\n    font-style: italic;\n  }\n  .pop-desc .pop-lead {\n    font-size: 1.05em;\n    border-left: 3px solid #1a1a1a;\n    padding-left: 1.1em;\n    margin-bottom: 1.75em;\n  }\n  .pop-desc h3 {\n    font-family: Georgia, 'Times New Roman', serif;\n    font-size: 0.8em;\n    font-weight: 700;\n    letter-spacing: 0.12em;\n    text-transform: uppercase;\n    color: #1a1a1a;\n    margin: 2em 0 0.75em 0;\n    padding-bottom: 0.4em;\n    border-bottom: 1px solid #d0d0d0;\n  }\n  .pop-desc ul.pop-chapters {\n    list-style: none;\n    padding: 0;\n    margin: 0 0 1.5em 0;\n  }\n  .pop-desc ul.pop-chapters li {\n    padding: 0.6em 0 0.6em 1.4em;\n    position: relative;\n    border-bottom: 1px solid #efefef;\n    font-size: 0.95em;\n  }\n  .pop-desc ul.pop-chapters li:last-child {\n    border-bottom: none;\n  }\n  .pop-desc ul.pop-chapters li::before {\n    content: '';\n    position: absolute;\n    left: 0;\n    top: 50%;\n    transform: translateY(-50%);\n    width: 5px;\n    height: 5px;\n    background: #1a1a1a;\n    border-radius: 50%;\n  }\n  .pop-desc .pop-note {\n    background: #f7f7f5;\n    border-left: 3px solid #1a1a1a;\n    padding: 1em 1.25em;\n    margin: 1.75em 0;\n    font-size: 0.93em;\n  }\n  .pop-desc .pop-note p {\n    margin: 0;\n  }\n  .pop-desc .pop-shipping {\n    display: flex;\n    flex-wrap: wrap;\n    gap: 0;\n    border: 1px solid #d0d0d0;\n    margin: 1.75em 0;\n    font-size: 0.8em;\n  }\n  .pop-desc .pop-shipping-item {\n    flex: 1 1 auto;\n    min-width: 0;\n    padding: 0.7em 0.5em;\n    text-align: center;\n    border-right: 1px solid #d0d0d0;\n    box-sizing: border-box;\n  }\n  .pop-desc .pop-shipping-item:last-child {\n    border-right: none;\n  }\n  .pop-desc .pop-shipping-item .pop-shipping-label {\n    display: block;\n    font-size: 0.75em;\n    letter-spacing: 0.08em;\n    text-transform: uppercase;\n    color: #777;\n    margin-bottom: 0.2em;\n    white-space: nowrap;\n  }\n  .pop-desc .pop-shipping-item .pop-shipping-value {\n    display: block;\n    font-weight: 700;\n    color: #1a1a1a;\n    word-break: break-word;\n  }\n  .pop-desc hr.pop-rule {\n    border: none;\n    border-top: 1px solid #d0d0d0;\n    margin: 2em 0;\n  }\n  .pop-desc .pop-author {\n    font-size: 0.93em;\n    color: #333;\n  }\n\u003c\/style\u003e\n\u003cdiv itemtype=\"https:\/\/schema.org\/Book\" itemscope=\"\" class=\"pop-desc\"\u003e\n\u003c!-- Plain SEO opener — keyword leads --\u003e\n\u003cp\u003e\u003cstrong itemprop=\"name\"\u003eA Comprehensive Introduction to Differential Geometry, Volume Four, Third Edition by Michael Spivak\u003c\/strong\u003e takes the classical surface theory of Volume Three into higher dimensions and codimensions — available here direct from \u003cspan itemprop=\"publisher\"\u003ePublish or Perish, Inc.\u003c\/span\u003e, the official publisher.\u003c\/p\u003e\n\u003cp class=\"pop-lead\"\u003eVolume Four contains Chapters 7, 8, and 9 of the unified work that Spivak regarded as spanning Volumes Three through Five. Its aim is to see how far the results of the previous volumes generalize when surfaces in three-dimensional Euclidean space are replaced by higher-dimensional manifolds of higher codimensions, immersed or imbedded in more general Riemannian manifolds. The volume also contains four substantial addenda to Chapter 7 — on the Laplacian, the Hodge star operator, isometric Riemannian manifolds, and better embedding invariants — which represent some of the deepest material in the series.\u003c\/p\u003e\n\u003chr class=\"pop-rule\"\u003e\n\u003ch3\u003eWhat This Volume Covers\u003c\/h3\u003e\n\u003cul class=\"pop-chapters\"\u003e\n\u003cli\u003e\n\u003cstrong\u003eChapter 7 — Higher Dimensions and Codimensions:\u003c\/strong\u003e The geometry of constant curvature manifolds including the standard models of spheres and hyperbolic spaces, stereographic projection, conformal models, and Beltrami's theorem. Curves in Riemannian manifolds via Frenet frames. The fundamental equations for submanifolds — the normal connection, Weingarten equations, Codazzi-Mainardi equations, Ricci equations, and the fundamental theorem for submanifolds of Euclidean and constant curvature manifolds. First consequences including the Theorema Egregium, mean curvature normal, umbilics, positive curvature and convexity. Further results on flat ruled surfaces and curves on hypersurfaces. Complete surfaces of constant curvature in spheres and hyperbolic space, the Hopf map, and Jorgens' theorem. Addenda cover the Laplacian, the Hodge star operator and Hodge's theorem, when two Riemannian manifolds are isometric, and better embedding invariants.\u003c\/li\u003e\n\u003cli\u003e\n\u003cstrong\u003eChapter 8 — The Second Variation:\u003c\/strong\u003e Two-parameter variations and the second variation formula. Jacobi fields and conjugate points. Minimizing and non-minimizing geodesics. The Hadamard-Cartan theorem. The Sturm Comparison theorem and Bonnet's theorem. Generalizations to higher dimensions including the Morse-Schoenberg comparison theorem, Meyer's theorem, and the Rauch comparison theorem. Synge's lemma and Synge's theorem. Cut points and Klingenberg's theorem.\u003c\/li\u003e\n\u003cli\u003e\n\u003cstrong\u003eChapter 9 — Variations of Length, Area, and Volume:\u003c\/strong\u003e Normal variations of surfaces in Euclidean three-space and minimal surfaces. Isothermal coordinates on minimal surfaces and Bernstein's theorem. The Weierstrass-Enneper representation. Associated minimal surfaces, Schwarz's theorem, Henneberg's minimal surface. Classical calculus of variations in n dimensions. The variation of volume formula and isoperimetric problems. Addenda on isothermal coordinates, immersed and imbedded surfaces with constant mean curvature, and the second variation of volume.\u003c\/li\u003e\n\u003c\/ul\u003e\n\u003chr class=\"pop-rule\"\u003e\n\u003cdiv class=\"pop-note\"\u003e\n\u003cp\u003eVolume Four requires the foundations of Volumes One, Two, and Three. It is the second of three volumes — Three, Four, and Five — that Spivak regarded as constituting a single unified work. Volume Five completes the series with the Gauss-Bonnet-Chern theorem and a comprehensive bibliography of the differential geometry literature. The complete five-volume set is also available at a 20% discount.\u003c\/p\u003e\n\u003c\/div\u003e\n\u003chr class=\"pop-rule\"\u003e\n\u003ch3\u003eAbout This Edition\u003c\/h3\u003e\n\u003cp\u003eEach 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.\u003c\/p\u003e\n\u003chr class=\"pop-rule\"\u003e\n\u003cdiv class=\"pop-shipping\"\u003e\n\u003cdiv class=\"pop-shipping-item\"\u003e\n\u003cspan class=\"pop-shipping-label\"\u003eShipping\u003c\/span\u003e \u003cspan class=\"pop-shipping-value\"\u003eFree FedEx 2Day (US)\u003c\/span\u003e\n\u003c\/div\u003e\n\u003cdiv class=\"pop-shipping-item\"\u003e\n\u003cspan class=\"pop-shipping-label\"\u003eProcessing\u003c\/span\u003e \u003cspan class=\"pop-shipping-value\"\u003e1 Business Day\u003c\/span\u003e\n\u003c\/div\u003e\n\u003cdiv class=\"pop-shipping-item\"\u003e\n\u003cspan class=\"pop-shipping-label\"\u003eDelivery\u003c\/span\u003e \u003cspan class=\"pop-shipping-value\"\u003e2 to 4 Days\u003c\/span\u003e\n\u003c\/div\u003e\n\u003cdiv class=\"pop-shipping-item\"\u003e\n\u003cspan class=\"pop-shipping-label\"\u003eReturns\u003c\/span\u003e \u003cspan class=\"pop-shipping-value\"\u003e30 Days\u003c\/span\u003e\n\u003c\/div\u003e\n\u003c\/div\u003e\n\u003chr class=\"pop-rule\"\u003e\n\u003ch3\u003eAbout the Author\u003c\/h3\u003e\n\u003cp itemprop=\"about\" class=\"pop-author\"\u003eMichael Spivak (1940–2020) earned his Ph.D. from Princeton University and is celebrated for writing mathematics textbooks of extraordinary rigor and clarity. \u003cem\u003eA Comprehensive Introduction to Differential Geometry\u003c\/em\u003e 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.\u003c\/p\u003e\n\u003c\/div\u003e","brand":"Publish or Perish, Inc. ®","offers":[{"title":"Default Title","offer_id":49983429574958,"sku":"POP-ACIDG-V4-3E","price":103.99,"currency_code":"USD","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0829\/3705\/3486\/files\/DG4covercrop.png?v=1763683772"},{"product_id":"a-comprehensive-introduction-to-differential-geometry-vol-5-3rd-edition","title":"A Comprehensive Introduction to Differential Geometry, Vol. 5, 3rd Edition by Michael Spivak | Publish or Perish","description":"\u003c!-- ============================================================\n     A COMPREHENSIVE INTRODUCTION TO DIFFERENTIAL GEOMETRY\n     VOLUME FIVE, THIRD EDITION\n     PRODUCT DESCRIPTION — Publish or Perish, Inc.\n     Paste into Shopify product description (HTML editor)\n     No emojis | No copyright symbols in body | Feed-safe\n\n     PAGE TITLE (SEO):\n     A Comprehensive Introduction to Differential Geometry, Vol. 5, 3rd Edition by Michael Spivak | Publish or Perish\n\n     META DESCRIPTION (157 characters — paste into SEO field):\n     A Comprehensive Introduction to Differential Geometry Vol. 5 by Michael Spivak — the Gauss-Bonnet-Chern theorem and series bibliography. Direct from the publisher. Free FedEx 2Day on US orders.\n     ============================================================ --\u003e\n\u003cstyle\u003e\n  .pop-desc {\n    font-family: Georgia, 'Times New Roman', serif;\n    color: #1a1a1a;\n    line-height: 1.75;\n    max-width: 720px;\n  }\n  .pop-desc p {\n    margin: 0 0 1.25em 0;\n    font-size: 1em;\n  }\n  .pop-desc strong {\n    font-weight: 700;\n  }\n  .pop-desc em {\n    font-style: italic;\n  }\n  .pop-desc .pop-lead {\n    font-size: 1.05em;\n    border-left: 3px solid #1a1a1a;\n    padding-left: 1.1em;\n    margin-bottom: 1.75em;\n  }\n  .pop-desc h3 {\n    font-family: Georgia, 'Times New Roman', serif;\n    font-size: 0.8em;\n    font-weight: 700;\n    letter-spacing: 0.12em;\n    text-transform: uppercase;\n    color: #1a1a1a;\n    margin: 2em 0 0.75em 0;\n    padding-bottom: 0.4em;\n    border-bottom: 1px solid #d0d0d0;\n  }\n  .pop-desc ul.pop-chapters {\n    list-style: none;\n    padding: 0;\n    margin: 0 0 1.5em 0;\n  }\n  .pop-desc ul.pop-chapters li {\n    padding: 0.6em 0 0.6em 1.4em;\n    position: relative;\n    border-bottom: 1px solid #efefef;\n    font-size: 0.95em;\n  }\n  .pop-desc ul.pop-chapters li:last-child {\n    border-bottom: none;\n  }\n  .pop-desc ul.pop-chapters li::before {\n    content: '';\n    position: absolute;\n    left: 0;\n    top: 50%;\n    transform: translateY(-50%);\n    width: 5px;\n    height: 5px;\n    background: #1a1a1a;\n    border-radius: 50%;\n  }\n  .pop-desc .pop-note {\n    background: #f7f7f5;\n    border-left: 3px solid #1a1a1a;\n    padding: 1em 1.25em;\n    margin: 1.75em 0;\n    font-size: 0.93em;\n  }\n  .pop-desc .pop-note p {\n    margin: 0 0 0.6em 0;\n  }\n  .pop-desc .pop-note p:last-child {\n    margin: 0;\n  }\n  .pop-desc .pop-shipping {\n    display: flex;\n    flex-wrap: wrap;\n    gap: 0;\n    border: 1px solid #d0d0d0;\n    margin: 1.75em 0;\n    font-size: 0.8em;\n  }\n  .pop-desc .pop-shipping-item {\n    flex: 1 1 auto;\n    min-width: 0;\n    padding: 0.7em 0.5em;\n    text-align: center;\n    border-right: 1px solid #d0d0d0;\n    box-sizing: border-box;\n  }\n  .pop-desc .pop-shipping-item:last-child {\n    border-right: none;\n  }\n  .pop-desc .pop-shipping-item .pop-shipping-label {\n    display: block;\n    font-size: 0.75em;\n    letter-spacing: 0.08em;\n    text-transform: uppercase;\n    color: #777;\n    margin-bottom: 0.2em;\n    white-space: nowrap;\n  }\n  .pop-desc .pop-shipping-item .pop-shipping-value {\n    display: block;\n    font-weight: 700;\n    color: #1a1a1a;\n    word-break: break-word;\n  }\n  .pop-desc hr.pop-rule {\n    border: none;\n    border-top: 1px solid #d0d0d0;\n    margin: 2em 0;\n  }\n  .pop-desc .pop-author {\n    font-size: 0.93em;\n    color: #333;\n  }\n\u003c\/style\u003e\n\u003cdiv itemtype=\"https:\/\/schema.org\/Book\" itemscope=\"\" class=\"pop-desc\"\u003e\n\u003c!-- Plain SEO opener — keyword leads --\u003e\n\u003cp\u003e\u003cstrong itemprop=\"name\"\u003eA Comprehensive Introduction to Differential Geometry, Volume Five, Third Edition by Michael Spivak\u003c\/strong\u003e 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 \u003cspan itemprop=\"publisher\"\u003ePublish or Perish, Inc.\u003c\/span\u003e, the official publisher.\u003c\/p\u003e\n\u003cp class=\"pop-lead\"\u003eVolume 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.\u003c\/p\u003e\n\u003chr class=\"pop-rule\"\u003e\n\u003ch3\u003eWhat This Volume Covers\u003c\/h3\u003e\n\u003cul class=\"pop-chapters\"\u003e\n\u003cli\u003e\n\u003cstrong\u003eChapter 10 — And Now a Brief Message from Our Sponsor (PDEs):\u003c\/strong\u003e 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.\u003c\/li\u003e\n\u003cli\u003e\n\u003cstrong\u003eChapter 11 — Existence and Non-Existence of Isometric Imbeddings:\u003c\/strong\u003e 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.\u003c\/li\u003e\n\u003cli\u003e\n\u003cstrong\u003eChapter 12 — Rigidity:\u003c\/strong\u003e 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.\u003c\/li\u003e\n\u003cli\u003e\n\u003cstrong\u003eChapter 13 — The Generalized Gauss-Bonnet Theorem and What It Means for Mankind:\u003c\/strong\u003e 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.\u003c\/li\u003e\n\u003c\/ul\u003e\n\u003chr class=\"pop-rule\"\u003e\n\u003cdiv class=\"pop-note\"\u003e\n\u003cp\u003e\u003cstrong\u003eThe Bibliography.\u003c\/strong\u003e 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.\u003c\/p\u003e\n\u003cp\u003eVolume Five requires the foundations of all four preceding volumes. The complete five-volume set is also available at a 20% discount.\u003c\/p\u003e\n\u003c\/div\u003e\n\u003chr class=\"pop-rule\"\u003e\n\u003ch3\u003eAbout This Edition\u003c\/h3\u003e\n\u003cp\u003eEach 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.\u003c\/p\u003e\n\u003chr class=\"pop-rule\"\u003e\n\u003cdiv class=\"pop-shipping\"\u003e\n\u003cdiv class=\"pop-shipping-item\"\u003e\n\u003cspan class=\"pop-shipping-label\"\u003eShipping\u003c\/span\u003e \u003cspan class=\"pop-shipping-value\"\u003eFree FedEx 2Day (US)\u003c\/span\u003e\n\u003c\/div\u003e\n\u003cdiv class=\"pop-shipping-item\"\u003e\n\u003cspan class=\"pop-shipping-label\"\u003eProcessing\u003c\/span\u003e \u003cspan class=\"pop-shipping-value\"\u003e1 Business Day\u003c\/span\u003e\n\u003c\/div\u003e\n\u003cdiv class=\"pop-shipping-item\"\u003e\n\u003cspan class=\"pop-shipping-label\"\u003eDelivery\u003c\/span\u003e \u003cspan class=\"pop-shipping-value\"\u003e2 to 4 Days\u003c\/span\u003e\n\u003c\/div\u003e\n\u003cdiv class=\"pop-shipping-item\"\u003e\n\u003cspan class=\"pop-shipping-label\"\u003eReturns\u003c\/span\u003e \u003cspan class=\"pop-shipping-value\"\u003e30 Days\u003c\/span\u003e\n\u003c\/div\u003e\n\u003c\/div\u003e\n\u003chr class=\"pop-rule\"\u003e\n\u003ch3\u003eAbout the Author\u003c\/h3\u003e\n\u003cp itemprop=\"about\" class=\"pop-author\"\u003eMichael Spivak (1940–2020) earned his Ph.D. from Princeton University and is celebrated for writing mathematics textbooks of extraordinary rigor and clarity. \u003cem\u003eA Comprehensive Introduction to Differential Geometry\u003c\/em\u003e 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.\u003c\/p\u003e\n\u003c\/div\u003e","brand":"Publish or Perish, Inc. ®","offers":[{"title":"Default Title","offer_id":49983410176302,"sku":"POP-ACIDG-V5-3E","price":112.99,"currency_code":"USD","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0829\/3705\/3486\/files\/Dg5covercrop.png?v=1763683772"},{"product_id":"mathpop-com-products-a-comprehensive-introduction-to-differential-geometry-complete-five-volume-set","title":"A Comprehensive Introduction to Differential Geometry: Complete Five-Volume Set by Michael Spivak | Publish or Perish","description":"\u003c!-- ============================================================\n     A COMPREHENSIVE INTRODUCTION TO DIFFERENTIAL GEOMETRY\n     COMPLETE FIVE-VOLUME SET\n     PRODUCT DESCRIPTION — Publish or Perish, Inc.\n     Paste into Shopify product description (HTML editor)\n     No emojis | No copyright symbols in body | Feed-safe\n\n     PAGE TITLE (SEO):\n     A Comprehensive Introduction to Differential Geometry: Five-Volume Set by Michael Spivak | Publish or Perish\n\n     META DESCRIPTION (157 characters — paste into SEO field):\n     The complete five-volume set of A Comprehensive Introduction to Differential Geometry by Michael Spivak — exclusive to mathpop.com — Free FedEx 2Day on US orders\n\n     URL HANDLE (update in Shopify):\n     differential-geometry-five-volume-set\n     ============================================================ --\u003e\n\n\u003cstyle\u003e\n  .pop-desc {\n    font-family: Georgia, 'Times New Roman', serif;\n    color: #1a1a1a;\n    line-height: 1.75;\n    max-width: 720px;\n  }\n  .pop-desc p {\n    margin: 0 0 1.25em 0;\n    font-size: 1em;\n  }\n  .pop-desc strong {\n    font-weight: 700;\n  }\n  .pop-desc em {\n    font-style: italic;\n  }\n  \/* Opening lead paragraph *\/\n  .pop-desc .pop-lead {\n    font-size: 1.05em;\n    border-left: 3px solid #1a1a1a;\n    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0.6em;\n    margin-bottom: 1.25em;\n    color: #1a1a1a;\n  }\n  \/* Author bio *\/\n  .pop-desc .pop-author {\n    font-size: 0.93em;\n    color: #333;\n  }\n\u003c\/style\u003e\n\n\u003cdiv class=\"pop-desc\" itemscope itemtype=\"https:\/\/schema.org\/Book\"\u003e\n\n  \u003c!-- Plain SEO opener — keyword leads --\u003e\n  \u003cp\u003e\u003cstrong itemprop=\"name\"\u003eA Comprehensive Introduction to Differential Geometry: Complete Five-Volume Set by Michael Spivak\u003c\/strong\u003e is the definitive work on the subject — a monumental series available exclusively at mathpop.com, direct from the official publisher, \u003cspan itemprop=\"publisher\"\u003ePublish or Perish, Inc.\u003c\/span\u003e\u003c\/p\u003e\n\n  \u003cp class=\"pop-lead\"\u003e\n    Spanning five hardcover volumes, this series builds a complete and rigorous foundation in differential geometry — from the basic theory of differentiable manifolds through the most advanced topics in the field. It is the standard reference for mathematicians and graduate students worldwide, and purchasing the complete set ensures you have the full arc of Spivak's treatment in hand.\n  \u003c\/p\u003e\n\n  \u003cspan class=\"pop-exclusive\"\u003eExclusive to mathpop.com\u003c\/span\u003e\n\n  \u003cdiv class=\"pop-discount\"\u003e\n    \u003cp\u003eBy purchasing the complete five-volume set you receive a \u003cstrong\u003e20% discount\u003c\/strong\u003e compared to buying the volumes individually.\u003c\/p\u003e\n  \u003c\/div\u003e\n\n  \u003chr class=\"pop-rule\"\u003e\n\n  \u003ch3\u003eVolume Overview\u003c\/h3\u003e\n\n  \u003cul class=\"pop-volumes\"\u003e\n    \u003cli\u003e\n      \u003cspan class=\"pop-vol-label\"\u003eVolume I\u003c\/span\u003e\n      \u003cstrong\u003eDifferentiable Manifolds\u003c\/strong\u003e — the foundational volume, covering the theory of differentiable manifolds, differential forms, the tangent bundle, vector fields, and integration of differential forms.\n    \u003c\/li\u003e\n    \u003cli\u003e\n      \u003cspan class=\"pop-vol-label\"\u003eVolume II\u003c\/span\u003e\n      \u003cstrong\u003eCurves, Surfaces, and Classical Geometry\u003c\/strong\u003e — begins with curves and surfaces in the plane and in space, then develops the foundational work of Gauss and Riemann including the birth of the Riemann curvature tensor.\n    \u003c\/li\u003e\n    \u003cli\u003e\n      \u003cspan class=\"pop-vol-label\"\u003eVolume III\u003c\/span\u003e\n      \u003cstrong\u003eCurvature\u003c\/strong\u003e — variations on the theme of curvature, classical surface theory, and the structures that connect the first two volumes to the advanced material that follows.\n    \u003c\/li\u003e\n    \u003cli\u003e\n      \u003cspan class=\"pop-vol-label\"\u003eVolume IV\u003c\/span\u003e\n      \u003cstrong\u003eHigher-Dimensional Geometry\u003c\/strong\u003e — higher-dimensional geometric structures including the Gauss-Bonnet-Chern theorem and its consequences.\n    \u003c\/li\u003e\n    \u003cli\u003e\n      \u003cspan class=\"pop-vol-label\"\u003eVolume V\u003c\/span\u003e\n      \u003cstrong\u003ePartial Differential Equations and Embeddings\u003c\/strong\u003e — isometric embeddings, rigidity, the Gauss-Bonnet theorem, first and quasi-linear PDEs, and the Cartan-Kahler theorem.\n    \u003c\/li\u003e\n  \u003c\/ul\u003e\n\n  \u003chr class=\"pop-rule\"\u003e\n\n  \u003ch3\u003eAbout This Edition\u003c\/h3\u003e\n\n  \u003cp\u003e\n    Each volume is hardcover with a matte laminate finish — a durable, clean presentation suited to a reference work of this kind. 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 — a meaningful quality in volumes used heavily for study and reference.\n  \u003c\/p\u003e\n\n  \u003cp\u003e\n    Each volume is bound using PUR (Polyurethane Reactive) adhesive — a high-performance binding method that forms a stronger, more flexible bond than traditional glues. PUR binding offers significantly greater page-pull strength, superior lay-flat quality, and reliable adhesion even under heavy use. It is the correct choice for a technical text of this density and size.\n  \u003c\/p\u003e\n\n  \u003chr class=\"pop-rule\"\u003e\n\n  \u003ch3\u003eWhy This Set\u003c\/h3\u003e\n\n  \u003cul class=\"pop-features\"\u003e\n    \u003cli\u003e\n\u003cstrong\u003eThe standard reference in the field\u003c\/strong\u003e — used and recommended by mathematicians and graduate programs worldwide for decades.\u003c\/li\u003e\n    \u003cli\u003e\n\u003cstrong\u003eComplete and self-contained\u003c\/strong\u003e — the five volumes form a unified treatment, best read and owned as a set.\u003c\/li\u003e\n    \u003cli\u003e\n\u003cstrong\u003eSpivak's celebrated clarity\u003c\/strong\u003e — rigorous without being inaccessible, with historical context that makes the development of the subject genuinely illuminating.\u003c\/li\u003e\n    \u003cli\u003e\n\u003cstrong\u003eHardcover throughout\u003c\/strong\u003e — all five volumes in durable hardcover, built for repeated reference use.\u003c\/li\u003e\n    \u003cli\u003e\n\u003cstrong\u003e20% set discount\u003c\/strong\u003e — the most economical way to acquire the complete series.\u003c\/li\u003e\n    \u003cli\u003e\n\u003cstrong\u003eDirect from the publisher\u003c\/strong\u003e — every copy guaranteed authentic, first-quality, and fulfilled by Publish or Perish, Inc.\u003c\/li\u003e\n  \u003c\/ul\u003e\n\n  \u003chr class=\"pop-rule\"\u003e\n\n  \u003c!-- Shipping strip --\u003e\n  \u003cdiv class=\"pop-shipping\"\u003e\n    \u003cdiv class=\"pop-shipping-item\"\u003e\n      \u003cspan class=\"pop-shipping-label\"\u003eShipping\u003c\/span\u003e\n      \u003cspan class=\"pop-shipping-value\"\u003eFree FedEx 2Day (US)\u003c\/span\u003e\n    \u003c\/div\u003e\n    \u003cdiv class=\"pop-shipping-item\"\u003e\n      \u003cspan class=\"pop-shipping-label\"\u003eProcessing\u003c\/span\u003e\n      \u003cspan class=\"pop-shipping-value\"\u003e1 Business Day\u003c\/span\u003e\n    \u003c\/div\u003e\n    \u003cdiv class=\"pop-shipping-item\"\u003e\n      \u003cspan class=\"pop-shipping-label\"\u003eDelivery\u003c\/span\u003e\n      \u003cspan class=\"pop-shipping-value\"\u003e2 to 4 Days\u003c\/span\u003e\n    \u003c\/div\u003e\n    \u003cdiv class=\"pop-shipping-item\"\u003e\n      \u003cspan class=\"pop-shipping-label\"\u003eReturns\u003c\/span\u003e\n      \u003cspan class=\"pop-shipping-value\"\u003e30 Days\u003c\/span\u003e\n    \u003c\/div\u003e\n  \u003c\/div\u003e\n\n  \u003chr class=\"pop-rule\"\u003e\n\n  \u003ch3\u003eAbout the Author\u003c\/h3\u003e\n\n  \u003cp class=\"pop-author\" itemprop=\"about\"\u003e\n    Michael Spivak (1940–2020) earned his Ph.D. from Princeton University and is celebrated for writing mathematics textbooks of extraordinary rigor and clarity. \u003cem\u003eA Comprehensive Introduction to Differential Geometry\u003c\/em\u003e 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.\n  \u003c\/p\u003e\n\n\u003c\/div\u003e","brand":"Publish or Perish, Inc. ®","offers":[{"title":"Default Title","offer_id":51974223626542,"sku":"DG-SET-1-5","price":451.95,"currency_code":"USD","in_stock":true}],"thumbnail_url":"\/\/cdn.shopify.com\/s\/files\/1\/0829\/3705\/3486\/files\/Gemini_Generated_Image_yzfjeryzfjeryzfj.png?v=1776749738"}],"url":"https:\/\/mathpop.com\/ca\/collections\/a-comprehensive-introduction-to-differential-geometry.oembed","provider":"Publish or Perish, Inc. ®","version":"1.0","type":"link"}