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A Comprehensive Introduction to Differential Geometry, Vol. II

Michael Spivak
Publish or Perish, Inc.
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The Basic Library List Committee strongly recommends this book for acquisition by undergraduate mathematics libraries.

[Reviewed by
Fernando Q. Gouvêa
, on

(Continued from the review of Volume I.)

In the second volume, Spivak begins to study the classical parts of differential geometry. He does just the right thing: assuming the language and background developed in the first volume, he goes through the material on curves and surfaces that one typically meets in a first elementary course.

The curves portion is pretty much what one would expect. This material is so straightforward that the new language really doesn't add much.

Once one starts talking about surfaces, however, Spivak really gets interesting. He takes a quasi-historical approach, meaning that he follows the historical order of progress when it is helpful and ignores it when it is not. After a brief chapter on the theory of surfaces before Gauss, what we get is a careful reading of Gauss's famous Disquisitiones Circa Superficies Curvas. (One of the innovations in this third edition is that it includes the full text, with Spivak's comments on facing pages. Hooray!) Jumping off from Gauss, the chapter reworks the theory in modern terms.

Then he does it again: starting from Riemann's famous essay "On the Hypotheses which lie at the Foundations of Geometry," Spivak develops the basic ideas in the geometry of manifolds, including a brilliant section called "The Birth of the Riemann Curvature Tensor." This is crucial: if one doesn't see how the curvature tensor is the natural way to generalize the notion of curvature, then all is lost.

The chapters that follow set up the rest of the basic structure of modern differential geometry. The subsequent volumes then apply all this: see the tables of contents of volumes three, four, and five (which the author says should be considered one multivolume book) for what is covered.

Throughout all this, Spivak maintains a sense of humor and a clear head. Chapter 10 in Volume five is called "And now a brief message from our sponsor," for example. It deals with partial differential equations. (I have heard rumors about yellow pigs, but cannot verify them.)

The books' presentation is charming: the covers are paintings in a primitive style that include references (some clear, some subtle) to each volume's mathematical content. (The joke about the Klein bottle from the 2nd edition is, alas, gone.) The fifth volume, appropriately, has a banner on the cover saying "All the way with Gauss-Bonnet." Sounds right to me.

Certainly, going "all the way" was an amazing achievement for the author, and any reader who manages to follow him through all five volumes will learn a lot, both about differential geometry and about the value of diligence. Few readers will do that, but many will read parts of this book, and those who do will profit from it.

There is simply no book comparable to this one, so no library should be without this five volume set. At the price, individuals may well want a set themselves, or at least the first two volumes. The books can be ordered from

Fernando Q. Gouvêa is professor of mathematics at Colby College.

  1. Curves in the Plane and in Space
        Curvature of plane curves. Convex curves. Curvature and torsion of
        space curves. The Serret-Frenet formulas. The natural from on a Lie group.
        Classification of plane curves under the group of special affine motions.
        Classification of curves in Euclidean n-space.

 2. What they knew about Surfaces before Gauss
        Euler's Theorem. Meusnier's Theorem.
 3. The Curvature of Surfaces in Space

            The Gauss map. Gaussian curvature. The Weingarten map; the first and
            second fundamental forms. The Theorema Egregium. Geodesics
            on a surface. The metric in geodesic polar coordinates. The integral
            of the curvature over a geodesic triangle.
            Addendum. The formula of Bertrand and Puiseux; Diquet's formula.

 4. The Curvature of Higher Dimensional Manifolds
            "On the Hypotheses which lie at the Foundations of Geometry"
            The form of the metric in Riemannian normal coordinates.
        C. A PRIZE ESSAY
            Necessary conditions for a metric to be flat. The Riemann curvature
            tensor. Sectional curvature. The Test Case; first version.
            Addendum. Finsler metrics.

 5. The Absolute Differential Calculus (The Ricci Calculus)
        Covariant derivatives. Ricci's Lemma. Ricci's identities. The curvature tensor.
        The Test Case; second version. Classical connections. The torsion tensor.
        Geodesics. Bianchi's identities.

 6. The Dell Operator
        Kozul connections. Covariant derivatives. Parallel translation. 
        The torsion tensor. The Levi-Civita connection. The curvature tensor.
        The Test Case; third version. Bianchi's identities. Geodesics. 
        The First Variation Formula.
        Addenda. Connections with the same geodesics. Riemann's
        invariant definition of the curvature tensor.

 7. The Repère Mobile (The Moving Frame)
        Moving frames. The structural equations of Euclidean space.
        The structural equations of a Riemannian manifold. The Test Case;
        fourth version. Adapted frames. The structural equations in polar
        coordinates. The Test Case; fifth version. The Test Case; sixth version.
        "The curvature determines the metric".  The 2-dimensional case.
        Cartan connections. Covariant derivatives and the torsion and curvature
        tensors. Bianchi's identities. 
        Addenda. Manifolds of constant curvature: Schur's Theorem; 
        The form of the metric in normal coordinates. Conformally equivalent manifolds.
        É. Cartan's treatment of normal coordinates.

 8. Connections in Principal Bundles
        Principal bundles. Lie groups acting on manifolds. A new definition of
        Cartan connections. Ehresmann connections. Lifts. Parallel translation
        and covariant derivatives. The covariant differential and the curvature
        form. The dual form and the torsion form. The structural equations.
        The torsion and curvature tensors. The Test Case; seventh version.
        Bianchi's identities. 
        Addenda. The tangent bundle of F(M). Complete connections.
        Connections in vector bundles. Flat connections.