A Comprehensive Introduction to Differential Geometry, Vol. 2, 3rd Edition [Hardcover]

Michael Spivak (Author), Michael Spivak (Author)

Product Details

• Hardcover
• Publisher: Publish or Perish; 3rd edition (January 1, 1999)
• Language: English
• ISBN-10: 0914098713
• ISBN-13: 978-0914098713
• Product Dimensions: 9.6 x 6.5 x 1 inches
• Shipping Weight: 1.8 pounds (View shipping rates and policies)
• Average Customer Review: 5.0 out of 5 stars  See all reviews (3 customer reviews)
• Amazon Best Sellers Rank: #434,022 in Books (See Top 100 in Books)

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56 of 57 people found the following review helpful:

5.0 out of 5 stars Wonderful exposition of the foundations of Curvature and Connections, October 4, 2005

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Paul Thurston (New York, NY USA) - See all my reviews
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This review is from: A Comprehensive Introduction to Differential Geometry, Vol. 2, 3rd Edition (Hardcover)

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.

0 of 21 people found the following review helpful:

5.0 out of 5 stars Long Journey - no damages, February 14, 2010

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Christoph Kneiidinger - See all my reviews
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This review is from: A Comprehensive Introduction to Differential Geometry, Vol. 2, 3rd Edition (Hardcover)

+: Product arrived without any damage.
+: Arrival date long before the promised, specified date
-: none

1 of 26 people found the following review helpful:

5.0 out of 5 stars A Comprehensive Introduction to Differential Geometry, Vol. 2, 3rd Edition, March 17, 2007

By 

beebsarelli "math dad" (dee troit) - See all my reviews

This review is from: A Comprehensive Introduction to Differential Geometry, Vol. 2, 3rd Edition (Hardcover)

Hours of reading fun! Well paced and twice the fun of Volume 1. Michael does it again! A spellbinding thriller from cover to cover. You gotta love it.