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Curves and singularities. A geometrical introduction to singularity theory. (English) Zbl 0534.58008
Cambridge etc.: Cambridge University Press. XII, 222 p. hbk: £25.00; $ 39.50; pbk: £8.95; $ 15.95 (1984).
This book is an excellent elementary introduction to the basic concepts of singularity theory, presented in their interplay with the classical differential geometry of plane and space (real) curves. Special emphasis is put on inflection points and vertices, envelopes on one-parameter families of curves and surfaces, evolutes, duals, caustics and generic properties of curves. A detailed discussion of a simple gravitational catastrophe machine (due to T. Poston), some interesting applications to geometrical optics (based on joint works of the authors with C. G. Gibson) and new connections of singularities with ordinary differential equations (continued recently by J. W. Bruce [Bull. Lond. Math. Soc. 16, 139-144 (1984; Zbl 0503.34003)] are also included in support of the usefulness of the singularity theory. The treatment is very accessible: the proofs are practically all complete to the least detail, the Sard theorem is given in an appendix and (simple variants of) the Thom transersality theorem is proved and used only in the last quarter of the book. The study of singularties (of functions) is essentially restricted to the one variable case, avoiding thus a lot of difficulties (e.g. more complicated normal forms, moduli). A lot of illuminating exercises are scattered throughout he book and detailed projects of further reading and investigations are also inserted for the interested/active student. The beautiful pictures (many of them computer made) and the inspiredly chosen mottos taken from Sherlock Holmes’s exploits hold the reader’s attention and help him to retain always the essential. The authors obviously enjoyed writing this book and the same joy is sure to pervade all the readers wishing to take their first steps into the wonderland of singularities.
Reviewer: A.Dimca

MSC:
58C25 Differentiable maps on manifolds
58K99 Theory of singularities and catastrophe theory
53A04 Curves in Euclidean and related spaces
58K35 Catastrophe theory
58-02 Research exposition (monographs, survey articles) pertaining to global analysis
53-02 Research exposition (monographs, survey articles) pertaining to differential geometry