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.
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.
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.
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.