Michael Spivak  Comprehensive Introduction to Differential Geometry
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Introduction to Differential Geometry  Volume two
Oct 23, 2000
Review by bnita
Rated a Very Helpful Review
Pros:clear, gradual introduction of the important notions, lots of pictures.
Cons:none
A Comprehensive Introduction to Differential Geometry" is a 5 volume book on Differential Geometry published for the first time in the seventies. In this review I intend to give a description of the second volume alone; the other volumes either have been already reviewed or they will follow shortly.
If you didn't already read my review about the first volume I suggest you do (http://www.epinions.com/bookreview302619A70D739F12294prod1); like any other book these volumes are related. Still I will try to give an independent opinion on this second volume and to make reference to the first one as little as possible.
Roughly speaking, the first volume described the manifolds and their differential properties (vectors, tensors and a little bit of calculus on these manifolds). This second volume is more geometric than the first one. First you learn about curves and surfaces in 2 and 3 dimensions in Euclidean space; then, same thing is presented in the light of manifolds. This leads to the notion of curvature, an essential one in Differential Geometry. The rest of the book is dedicated to it (properties, identities, special cases, etc.). As usual I will give a brief description of each one of the chapters.
Chapter one  Curves in the Plane and in Space: This chapter is concerned with curves in an Euclidean space (R^2 or R^3). You will have a chance to remember the things you already knew from Calculus and to learn new stuff; everything is presented with an eye on the curvature and torsion notions. Again I was surprised by the clarity of the text and by the abundance of pictures. There is an average of two pictures on every page! And that's a very good thing when you learn geometry.
Chapter two  What they knew about Surfaces before Gauss: This is the shortest chapter of the book (I guess they didn't know too much before Gauss; by the way there is an interesting story about Gauss at the end of the review). It contains two theorems Euler's and Meusnier's with proofs, pictures and explanations.
Chapter three  The curvature of Surfaces in Space: This chapter goes deep in the theory of surfaces. The Gauss map is introduced and it leads smoothly to the notion of Gaussian curvature (this is the one that gives us that the plane has a zero curvature and the sphere has a positive curvature equal to 1/r^2 where r is it's radius). The fundamental forms are presented with the use of Weingarten map. Also you will have a chance to calculate the geodesics of some of the most known surfaces: plane, sphere, cylinder etc.
Chapter four  The curvature of the higher dimensional manifolds: This chapter has a very interesting and unique structure. It starts with a lecture given by Riemann on October 10, 1854 at Gottingen University. It was one of the most important events in Differential Geometry. At first it looks impossible to follow, but in the next section the author explains every little notion of that lecture and show how Riemann's ideas inspired manifolds and the tensors that is named after him (Riemann's curvature tensor).
Chapter five  The absolute differential calculus: All the calculus we've done so far was, in a way or another local. the defined derivatives usually depended on tangent vectors. In this chapter the covariant (or absolute) derivatives are defined and used to obtain Ricci's identities, the torsion tensor and Bianchi identities.
Chapter six  The dell operator: This section presents some of the most important connections (derivatives) on manifolds, Koszul and LeviCivita. The notion of parallel translation (or transport) and how to explain the torsion and the curvature tensors in terms of it is also in this chapter.
Chapter seven  The moving frame: The structural equations for the Euclidean space introduce the ones for Riemannian manifolds (Cartan's). These are also given in some adapted frames as well as in polar coordinates. The chapter has also some short presentations on some subjects of interest: Manifolds of Constant Curvature, Conformally equivalent manifolds, Normal Coordinates.
Chapter eight  Connections in principal bundles: An important derivative is presented in this chapter namely the Lie derivative. It is introduced by means of the principal bundles. Covariant differential, curvature form, structural equations and the torsion and curvature tensor are also presented from this point of view.
Conclusions: This second volume goes deep in the differential properties of the manifolds. The most important things to learn from it are the curvature tensor and the connections on manifolds. Again I was amazed by the clarity of writing and by the numerous examples and pictures. There are no exercises in this volume.
Short story about Gauss 17771855: Some mathematicians thing Gauss was the greatest mathematician ever. Some think he was a fraud. But at least everybody accepts the fact that he was the most well know mathematician in his life time. A lot of young mathematicians wrote to him saying that they thought they discovered some new things and would like Gauss to take a look at them. Gauss would read the manuscript and reply either that it's useless or that he also looked at these kind of problems and found such and such. Some say that he was just stealing the poor guys work; some other say that he was just brilliant and knew a lot. Some of the things attributed to him are: the theory of curves, the fundamental theorem of Algebra, curved geometry. But, as the X files always say, The truth is out there.
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