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Synthetic geometry of manifolds. (English) Zbl 1210.51001
Cambridge Tracts in Mathematics 180. Cambridge: Cambridge University Press (ISBN 978-0-521-11673-2/hbk). xiii, 302 p. (2010).
This book is intended as an introduction to synthetic differential geometry. For the reader who needs such an introduction, synthetic differential geometry can be thought of (very informally) as a “third way” for calculus. Making calculus rigorous is largely a matter of dealing properly with the engineering student’s useful but nebulous idea of “infinitesimal quantity”. The classical limit formalism dances around it. Nonstandard analysis shows, via logic, that something very much like it (including both the utility and the nebulosity) can consistently be introduced to the real numbers. Synthetic differential geometry also uses a coordinate ring with infinitesimal elements, but allows them to have a cleaner structure; those elements are actually nilpotent. (However, there are surprises; the set of nilpotent elements need not be closed under addition.) Also, synthetic differential geometry allows far more freedom to choose the axioms that the spaces being studied should satisfy.
The book covers, from a synthetic viewpoint, a wide range of topics familiar from of classical (local, finite-dimensional) differential geometry. The concepts of “atlas” and “chart” are much as one would expect; but a manifold inherits from the coordinate ring, via these charts, a symmetric, reflexive (but typically non-transitive) “neighbor” relation. Thus, a synthetic manifold has a nontrivial graph structure; and many ideas such as “distribution” or “differential form”, can be interpreted as discrete combinatorial concepts within this.
The book would certainly make a good graduate textbook. It is clearly written and contains a reasonable number of nontrivial exercises. The introduction claims that it requires only a background in multivariate calculus, linear algebra, commutative ring theory, and “basic category theory”. This (as usual) is somewhat optimistic; while the first three sections of the appendix do define topoi, models, and sheaf semantics, they are very brief. The reader who has seen these concepts before will have a much easier time – but of course most will have.
Reviewer’s remark: A couple of minor quibbles: the term “parallelepipedum” should surely be “parallelepipedon”. I also found one typographical error: the “\(z\)” on page 60, line 6 should be an “\(x\)”.

51-02 Research exposition (monographs, survey articles) pertaining to geometry
51K10 Synthetic differential geometry