Barrett O'Neill's introduction to differential geometry was first published in 1966. At that point, it pretty much had the field to itself. There was simply no other book taking a modern approach to the material that could be used as an undergraduate text. Forty years later, O'Neill has lots of competition, of course. But it's still a beautiful book, well-written and displaying a deep understanding of the subject. And with old-fashioned artist-drawn illustrations. It's good to see it come back in a new edition.
The changes to the text are fairly minimal. There are lots of corrections, and the exercises have been significantly improved. Beyond that, the only real change is the addition of a very useful appendix showing how to use Mathematica or Maple to do the more important computations in elementary differential geometry. As the author points out, it is instructive to do these calculations by hand once or twice, but once the matter is understood one should use technology.
For instructors looking for an undergraduate textbook, the most problematic aspect of this book will be O'Neill's decision to use the language of tangent vectors, differential forms, and covariant derivatives. There is no doubt that this is the right language, but using it in an introductory book means quite a large amount of setup is needed right at the beginning of the course. This can get in the way of the geometry, as I learned when I tried to teach from this book. On the other hand, this is the language used in more advanced books, so learning it can be very useful to the student.
O'Neill has remained alone in choosing this approach. His competitors (e.g., the books by do Carmo, Oprea, and Pressley) all prefer to avoid differential forms. I don't know which I prefer; it depends on the day. But I do know that O'Neill's book is very nice, the sort of book that can make the right student learn to love the subject.
Fernando Q. Gouvêa read parts of O'Neill's book way back when he was an undergraduate. Nowadays he's an old fogey who teaches at Colby College in Waterville, ME.
Chapter 1: Calculus on Euclidean Space:
Euclidean Space. Tangent Vectors. Directional Derivatives. Curves in R3. 1-forms. Differential Forms. Mappings.
Chapter 2: Frame Fields:
Dot Product. Curves. The Frenet Formulas. ArbitrarySpeed Curves. Covariant Derivatives. Frame Fields. Connection Forms. The Structural Equations.
Chapter 3: Euclidean Geometry:
Isometries of R3. The Tangent Map of an Isometry. Orientation. Euclidean Geometry. Congruence of Curves.
Chapter 4: Calculus on a Surface:
Surfaces in R3. Patch Computations. Differentiable Functions and Tangent Vectors. Differential Forms on a Surface. Mappings of Surfaces. Integration of Forms. Topological Properties. Manifolds.
Chapter 5: Shape Operators:
The Shape Operator of M R3. Normal Curvature. Gaussian Curvature. Computational Techniques. The Implicit Case. Special Curves in a Surface. Surfaces of Revolution.
Chapter 6: Geometry of Surfaces in R3:
The Fundamental Equations. Form Computations. Some Global Theorems. Isometries and Local Isometries. Intrinsic Geometry of Surfaces in R3. Orthogonal Coordinates. Integration and Orientation. Total Curvature. Congruence of Surfaces.
Chapter 7: Riemannian Geometry: Geometric Surfaces. Gaussian Curvature. Covariant Derivative. Geodesics. Clairaut Parametrizations. The Gauss-Bonnet Theorem. Applications of Gauss-Bonnet.
Chapter 8: Global Structures of Surfaces: Length-Minimizing Properties of Geodesics. Complete Surfaces. Curvature and Conjugate Points. Covering Surfaces. Mappings that Preserve Inner Products. Surfaces of Constant Curvature. Theorems of Bonnet and Hadamard.
Answers to Odd-Numbered Exercises