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Differential Geometry of Curves and Surfaces: A Concise Guide

Victor A. Toponogov with the editorial assistance of Vladimir Y. Rovenski
Publication Date: 
Number of Pages: 
[Reviewed by
Fernando Q. Gouvêa
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Toponogov's "concise guide" to elementary differential geometry has the potential to be a useful reference and/or review book, but I wouldn't want to learn the subject from it. The dominant word is, indeed, "concise." As I read through it, I repeatedly had the same experience: when the text dealt with material I already knew well, I could follow without problems; when it dealt with things I remembered vaguely, it was also helpful; but the bits I had not met before were impenetrable.

This seems to be a set of course notes, with little alteration. The English is occasionally a little bit strange (there is no indication of a Russian edition, so I presume the English is either the author's or the editor's). There is very little non-mathematical talk, however, so this doesn't get in the way too much. Still, it is sometimes confusing. For example, on page 152, we have "Later on, we will agree... [to use the Einstein summation convention.]" I think what is meant is "from now on," rather than "later on." Otherwise, I don't know what the equations at the bottom of that very same page mean!

[I should take a time out here to protest the use of the Einstein summation convention, perhaps the most unilluminating notational choice ever. We hates it!]

Perhaps the best contribution of this little book lies in some of the examples (usually labeled "problems") and in the effort to give geometric content to some of the standard ideas. But one definitely shouldn't start here.

Fernando Q. Gouvêa is professor of mathematics at Colby College in Waterville, ME, where he has taught elementary differential geometry several times.

 Chapter 1 Curves in a 3-dimensional Euclidean space and in the plane: Preliminaries.- Definition and methods of curves presentation.- Tangent line and an osculating plane.- Length of a curve.- Problems: plane convex curves.- Curvature of a curve.- Problems: curvature of plance curves.- Torsion of a curve.- Frenet formulas and the natural equation of a curve.- Problems: space curves- Phase length of a curve and Fenchel-Reshetnyak inequality.- Exercises Chapter 2 Extrinsic geometry of surfaces in a 3-dimensional Euclidean space.- Definition and methods of generating surfaces.- Tangent plane.- First fundamental form of a surface.- Second fundamental form of a surface.- The third fundamental form of a surface.- Classes of surfaces.- Some classes of curves on a surface.- The main equations of the surfaces theory.- Appendix: Indicatrix of a surface of revolution.- Exercises Chapter 3 Intrinsic geometry of surfaces.- Introducing notions.-Covariant derivative of a vector field.- Parallel translation of a vector along a curve on a surface.- Geodesics.- Shortest paths and geodesics.- Special coordinate system.- Gauss-Bonet theorem and comparison theorem for the angles of a triangle.- Local comparison theorems for triangle.- Alexandrov comparison theorem for the angles of a triangle.- Problems.- Bibliography.- Index