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.