--- product_id: 8855222 title: "A Comprehensive Introduction to Differential Geometry, Vol. 2, 3rd Edition" price: "COP 831028" currency: COP in_stock: true reviews_count: 12 url: https://www.desertcart.co/products/8855222-a-comprehensive-introduction-to-differential-geometry-vol-2-3rd-edition store_origin: CO region: Colombia --- # A Comprehensive Introduction to Differential Geometry, Vol. 2, 3rd Edition **Price:** COP 831028 **Availability:** ✅ In Stock ## Quick Answers - **What is this?** A Comprehensive Introduction to Differential Geometry, Vol. 2, 3rd Edition - **How much does it cost?** COP 831028 with free shipping - **Is it available?** Yes, in stock and ready to ship - **Where can I buy it?** [www.desertcart.co](https://www.desertcart.co/products/8855222-a-comprehensive-introduction-to-differential-geometry-vol-2-3rd-edition) ## Best For - Customers looking for quality international products ## Why This Product - Free international shipping included - Worldwide delivery with tracking - 15-day hassle-free returns ## Description A Comprehensive Introduction to Differential Geometry, Volume Two, Third Edition" by Michael Spivak, published by Publish or Perish, Inc. in Houston, Texas, 2025, covers advanced topics in differential geometry. The book follows a semi-historical path, focusing on classical differential geometry in Chapters 3 and 4. Although it lacks problem sets, it compensates with a comprehensive bibliography in the final volume. The book delves into intricate details of differential geometry, providing a thorough understanding of the subject through historical context, theoretical foundations, and advanced mathematical formulations. Key topics include curves in the plane and in space, the curvature of surfaces in space, the curvature of higher-dimensional manifolds, the absolute differential calculus (Ricci calculus), the r operator, the repère mobile (moving frame), and connections in principal bundles. The document section is an excerpt from the book, specifically discussing Gauss' investigations of curved surfaces. It provides detailed explanations and proofs of various geometric properties and theorems related to the curvature of surfaces. The section combines historical context, mathematical rigor, and detailed proofs to explore the fundamental aspects of differential geometry, focusing on Gauss' contributions and Riemann's advancements. Overall, the book is a valuable resource for anyone looking to deepen their understanding of differential geometry, offering a comprehensive and authoritative guide to the subject. Review: Wonderful exposition of the foundations of Curvature and Connections - This book is the second volume of the 3rd edition in a five volume series on differential geometry. The focus here is on the foundations of curvature and connections. The only prerequisite for volume II is a careful study of volume I. In particular, you'll need a good understanding of the Riemannian metric and you'll need to be comfortable with manipulating differential forms. Also pay attention to the differential equations material used to establish Frobenius Integrability in Chapter 6 of volume I. In addition, you'll need the main concepts from the Lie Groups study of Chapter 10 of volume I. The author begins the study of curvature with a review of the classical theory of curvature of curves and surfaces in Chapters 1 and 2. These chapters are written in style that helps the reader anticipate more general results for Riemannian manifolds. For example, the reader will notice the rotation index of a planar curve can be represented in terms of its total curvature; a result which foreshadows the Gauss-Bonnet Theorem. Both Euler's Theorem and Meusnier's Theorem for surfaces embedded in Euclidean 3-space are studied. Chapter 3 details the geometry of surfaces as developed by Gauss. Spivak's treatment here is very unusual, and, in Part A of this chapter, the author actually gives an English translation of original paper of Gauss. Reading this is a bit unusual as the author alternates the translation of Gauss on a page with comments by the author on the preceding page. Part B of the chapter gives the accounting of the Gauss Theory in modern notion. Part B is delightfully geometric and includes all of the 'greatest hits' from the theory, including the Theorema Egreguim and the Triangle Excess Theorem. Chapter 4 studies Riemann's theory of curvature of manifolds, and contains 4 parts. Part A and Part C are English translations of Riemann's foundational work, while Part B and Part D cast this work in the light of more modern notion. Riemann's curvature tensor is built up from an intuitive study of the second-order terms in the Taylor series expansion of the Riemannian metric. The author also introduces what he calls the "Test Case" for curvature theory: Flat manifolds are locally isometric to Euclidean space. Spivak uses this "Test Case" repeatedly throughout the remainder of the text to reinforce the various notion of curvature as he studies the work of Riemann, Ricci, Kozul, Cartan and Ehresmann. Chapter 5 (the Debauch of Indices) studies the work of Christoffel and Ricci in developing the covariant derivative. The aim of this work is to simplify the somewhat cumbersome formulas for Riemann's curvature tensor. The reader quickly sees that effort, called absolute differential calculus, is not altogether successful and leads to an veritable explosion of multi-indexed quantities and even harder-to-penetrate formulas. Clearly a better way is needed if we are to move forward with our study of differential geometry. The "way forward" is Kozul's concept of the connection and this is introduced in Chapter 6. First, note that the connection here is one of the versions of the introduced by Kozul as a map of pairs of vector fields to a vector field. Another useful version, not studied in volume II, is to consider the connection as a Hessian which maps any smooth function to a bilinear form on the tangent space. Second, note that Chapter 6 is usually the starting point for most treatments of curvature in differential geometry (e.g Do Carmo's "Riemannian Geometry"). Without the motivating material from the previous chapters, it would be difficult to understand the need for(or the point of) Kozul's connection. Cartan's theory of curvature via a study of moving frames is detailed in Chapter 7. The author is careful to intuitively motivate Cartan's deviation from Euclidean concept as represented in the structure equations. Cartan's curvature tensor is shown to agree with Riemann's tensor, the "Test Case" is revisited, and the well-known fact that the curvature determines the Riemannian metric is established. Building on the orthonormal frames from the previous chapter, Spivak now considers Ehresmann's theory of connections in principal bundles in Chapter 8. The main results here introduce the Ehresmann connection on the frame bundle, and gives the Kozul connection as a Lie derivative, thought of as the Cartan connection obtained from the Ehresmann connection. My only complaint is that the author didn't include any exercises in this second volume. This is a real shame as the exercises in the first volume were very well-designed and one of the highlights of that text. Review: a classic - This was one of the books that helped me decide to get a phd in math (even though I didn't officially study differential geometry). Spivak's books read like chalkboard lectures by a superb lecturer. The treatment of curvature of curves and surfaces in the first two chapters are really good. I am still absorbing the meat of the book (Riemannian geometry) and I have been reading it on and off for 20 years. ## Features - Focus is on the foundations of geometry of curves and intrinsic geometry of surfaces, connections for tensor bundles and general fibre bundles. ## Technical Specifications | Specification | Value | |---------------|-------| | Best Sellers Rank | #1,270,111 in Books ( See Top 100 in Books ) #99 in Differential Geometry (Books) #3,465 in Mathematics (Books) | | Customer Reviews | 4.7 out of 5 stars 39 Reviews | ## Images ![A Comprehensive Introduction to Differential Geometry, Vol. 2, 3rd Edition - Image 1](https://m.media-amazon.com/images/I/61y7zV0hQvL.jpg) ## Customer Reviews ### ⭐⭐⭐⭐⭐ Wonderful exposition of the foundations of Curvature and Connections *by P***N on October 4, 2005* This book is the second volume of the 3rd edition in a five volume series on differential geometry. The focus here is on the foundations of curvature and connections. The only prerequisite for volume II is a careful study of volume I. In particular, you'll need a good understanding of the Riemannian metric and you'll need to be comfortable with manipulating differential forms. Also pay attention to the differential equations material used to establish Frobenius Integrability in Chapter 6 of volume I. In addition, you'll need the main concepts from the Lie Groups study of Chapter 10 of volume I. The author begins the study of curvature with a review of the classical theory of curvature of curves and surfaces in Chapters 1 and 2. These chapters are written in style that helps the reader anticipate more general results for Riemannian manifolds. For example, the reader will notice the rotation index of a planar curve can be represented in terms of its total curvature; a result which foreshadows the Gauss-Bonnet Theorem. Both Euler's Theorem and Meusnier's Theorem for surfaces embedded in Euclidean 3-space are studied. Chapter 3 details the geometry of surfaces as developed by Gauss. Spivak's treatment here is very unusual, and, in Part A of this chapter, the author actually gives an English translation of original paper of Gauss. Reading this is a bit unusual as the author alternates the translation of Gauss on a page with comments by the author on the preceding page. Part B of the chapter gives the accounting of the Gauss Theory in modern notion. Part B is delightfully geometric and includes all of the 'greatest hits' from the theory, including the Theorema Egreguim and the Triangle Excess Theorem. Chapter 4 studies Riemann's theory of curvature of manifolds, and contains 4 parts. Part A and Part C are English translations of Riemann's foundational work, while Part B and Part D cast this work in the light of more modern notion. Riemann's curvature tensor is built up from an intuitive study of the second-order terms in the Taylor series expansion of the Riemannian metric. The author also introduces what he calls the "Test Case" for curvature theory: Flat manifolds are locally isometric to Euclidean space. Spivak uses this "Test Case" repeatedly throughout the remainder of the text to reinforce the various notion of curvature as he studies the work of Riemann, Ricci, Kozul, Cartan and Ehresmann. Chapter 5 (the Debauch of Indices) studies the work of Christoffel and Ricci in developing the covariant derivative. The aim of this work is to simplify the somewhat cumbersome formulas for Riemann's curvature tensor. The reader quickly sees that effort, called absolute differential calculus, is not altogether successful and leads to an veritable explosion of multi-indexed quantities and even harder-to-penetrate formulas. Clearly a better way is needed if we are to move forward with our study of differential geometry. The "way forward" is Kozul's concept of the connection and this is introduced in Chapter 6. First, note that the connection here is one of the versions of the introduced by Kozul as a map of pairs of vector fields to a vector field. Another useful version, not studied in volume II, is to consider the connection as a Hessian which maps any smooth function to a bilinear form on the tangent space. Second, note that Chapter 6 is usually the starting point for most treatments of curvature in differential geometry (e.g Do Carmo's "Riemannian Geometry"). Without the motivating material from the previous chapters, it would be difficult to understand the need for(or the point of) Kozul's connection. Cartan's theory of curvature via a study of moving frames is detailed in Chapter 7. The author is careful to intuitively motivate Cartan's deviation from Euclidean concept as represented in the structure equations. Cartan's curvature tensor is shown to agree with Riemann's tensor, the "Test Case" is revisited, and the well-known fact that the curvature determines the Riemannian metric is established. Building on the orthonormal frames from the previous chapter, Spivak now considers Ehresmann's theory of connections in principal bundles in Chapter 8. The main results here introduce the Ehresmann connection on the frame bundle, and gives the Kozul connection as a Lie derivative, thought of as the Cartan connection obtained from the Ehresmann connection. My only complaint is that the author didn't include any exercises in this second volume. This is a real shame as the exercises in the first volume were very well-designed and one of the highlights of that text. ### ⭐⭐⭐⭐⭐ a classic *by E***Z on May 11, 2019* This was one of the books that helped me decide to get a phd in math (even though I didn't officially study differential geometry). Spivak's books read like chalkboard lectures by a superb lecturer. The treatment of curvature of curves and surfaces in the first two chapters are really good. I am still absorbing the meat of the book (Riemannian geometry) and I have been reading it on and off for 20 years. ### ⭐⭐⭐⭐⭐ What can one say about Spivak's books on differential geometry ... *by C***R on April 18, 2015* What can one say about Spivak's books on differential geometry. I used a complete set in my undergrad years and used them so much that I wanted a new copy! Definitely one set of books worth having in every mathematician's library. ## Frequently Bought Together - A Comprehensive Introduction to Differential Geometry, Vol. 2, 3rd Edition - A Comprehensive Introduction to Differential Geometry, Vol. 1, 3rd Edition - A Comprehensive Introduction to Differential Geometry, Vol. 4, 3rd Edition --- ## Why Shop on Desertcart? - 🛒 **Trusted by 1.3+ Million Shoppers** — Serving international shoppers since 2016 - 🌍 **Shop Globally** — Access 737+ million products across 21 categories - 💰 **No Hidden Fees** — All customs, duties, and taxes included in the price - 🔄 **15-Day Free Returns** — Hassle-free returns (30 days for PRO members) - 🔒 **Secure Payments** — Trusted payment options with buyer protection - ⭐ **TrustPilot Rated 4.5/5** — Based on 8,000+ happy customer reviews **Shop now:** [https://www.desertcart.co/products/8855222-a-comprehensive-introduction-to-differential-geometry-vol-2-3rd-edition](https://www.desertcart.co/products/8855222-a-comprehensive-introduction-to-differential-geometry-vol-2-3rd-edition) --- *Product available on Desertcart Colombia* *Store origin: CO* *Last updated: 2026-05-28*