{"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","url":"https:\/\/mathpop.com\/ca\/products\/a-comprehensive-introduction-to-differential-geometry-vol-2-3rd-edition","provider":"Publish or Perish, Inc. ®","version":"1.0","type":"link"}