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A Comprehensive Introduction to Differential Geometry, Vol. I

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

In the first line of his introduction to the first edition of this book, Michael Spivak says that "For many years I have wanted to write the Great American Differential Geometry book." And he did. The brashness of youth must have helped, since the book grew to be five volumes long. In his introduction to the revised third edition, he says:

At one time I had optimistically planned to completely revise all this material for the momentous occasion [when the notes became a book], but I soon realized the futility of such an undertaking. As I examined these five volumes, written so many years ago, I could scarcely believe that I had once had the energy to learn so much material, or even recall how I had unearthed some of it.

The main change in the third edition is that the book has been completely re-set using TeX. (The pictures have also been redrawn.) The original edition had been simply a bound set of notes (on, I think, standard 8.5x11 paper). The second edition was more like a real book, but it was still simply photographed from typed notes. This version is much more pleasant to read.

I first met these books shortly after my first course in differential geometry. The professor followed the standard approach for such courses: minimize the machinery needed by restricting the material to curves and surfaces. This worked well, but it meant that when I wanted to move on to more advanced material, I found the language well-nigh impenetrable. What did tensors and connections have to do with the Frenet equations and the first and second fundamental forms? That, plus the horrible notation in which most books were written (for example, the nasty "Einstein summation convention"), effectively left me out in the cold.

I had read Spivak's Calculus on Manifolds, so when I discovered the first volume of his Comprehensive Introduction I was happy to give it a try. It was hard going. I read most of it, absorbed at least some of it, and learned to respect the author and his achievement.

In this first volume, Spivak sets up the language and context for the rest of the book, developing the theory of differentiable manifolds, vector fields, differential forms, and Riemannian metrics, with side looks at Lie groups and algebraic topology. For someone not already interested in the subject, that is, of course, overkill, and very hard to motivate. But "comprehensive" is the driving word, and for those who are already motivated to learn this language, this is one of the best places in which to do that.

The geometry proper starts in Volume II.

Fernando Q. Gouvêa is professor of mathematics at Colby College in Waterville, ME.

 1. Manifolds
         Elementary properties of manifolds. Examples of manifolds.
 2. Differential Structures
        Smooth structures. Smooth functions. Partial derivatives. Critical points.
        Immersion theorems. Partitions of unity.

 3. The Tangent Bundle
         The tangent space of Euclidean space. The tangents space of an imbedded
        manifold. Vector bundles. The tangent bundle of a manifold. Equivalence
        classes of curves, and derivations. Vector fields. Orientation.
        Addendum: Equivalence of tangent bundles.
 4. Tensors
        The dual bundle. The differential of a function. Classical versus modern
        terminology. Multilinear functions. Covariant and contravariant tensors.
        Mixed tensors, and contraction.

 5. Vector Field and Differential Equations
        Integral curves. Existence and uniqueness theorems. The local flow.
        One-parameter groups of diffeomorphisms. Lie derivatives. Brackets.
        Addenda: Differential equations. Parameter curves in two dimensions.

 6. Integral manifolds
        Prologue; classical integrability theorems. Local Theory; Frobenius
        integrability theorem. Global Theory.

 7. Differential Forms
        Alternating functions. The wedge product. Forms. Differential of a form.
        Frobenius integrability theorem (second version). Closed and exact forms.
        The Poincaré Lemma.

 8. Integration
        Classical line and surface integrals. Integrals over singular cubes.
        The boundary of a chain. Stokes' Theorem. Integrals over manifolds.
        Volume elements. Stokes' Theorem. de Rham cohomology.

 9. Riemannian Metrics
        Inner products. Riemannian metrics. Length of curves. The calculus of
        variations. The First Variation Formula and geodesics. The exponential

map. Geodesic completeness. 
        Addendum: Tubular neighborhoods.

10. Lie Groups
         Lie groups. Left invariant vector fields. Lie algebra. Subgroups and
        subalgebras. Homomorphisms. One-parameter subgroups.
        The exponential map. Closed subgroups. Left invariant forms.
        Bi-invariant metrics. The equations of structure.

11. Excursion in the Realm of Algebraic Topology
        Complexes and exact sequences. The Mayer-Vietoris sequence.
        Triangulations. The Euler characteristic. Mayer-Vietoris sequence
        for compact supports. The exact sequence of a pair. Poincaré Duality.
        The Thom class. Index of a vector field.  Poincaré-Hopf  Theorem.