{"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","url":"https:\/\/mathpop.com\/ca\/products\/a-comprehensive-introduction-to-differential-geometry-vol-4-3rd-edition","provider":"Publish or Perish, Inc. ®","version":"1.0","type":"link"}