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I am seeking good books on the geometry of surfaces in Euclidean space, which would in particular discuss Darboux frames. Please explain for each suggestion why you like this book (classics are welcome).

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Chern explains moving frames in a masterful 4 pages: (1990, pp. 210–213).

Further nice book-length treatments using them:

  • Valiron (1950, Chap XIII–XIV) — last of the classic Cours, translated, very complete.

  • Guggenheimer (1963, Chap. 10–11) — actually says “Darboux frames” throughout.

  • O’Neill (1966, Chap. VI–VII) — standard choice.

  • Cartan (1967, Chap. III) — as learned from his father(?)

  • Do Carmo (1971, Chap. 5–6) — not (1976) which mostly does without frames.

  • Blaschke (1973, Chap. 5–6) — as taught to Chern(?)

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There is a very general discussion of moving frames in Clelland's book, From Frenet to Cartan: The Method of Moving Frames, and this in particular includes discussion of Darboux's use of frames on surfaces. Darboux's own 4 volume Lecons sur la Theorie Generale des Surfaces at les Applications Geometriques du Calcul Infinitesimal is quite readable, although very long.

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  • $\begingroup$ You could also look at my notes, appendix D on moving frames: arxiv.org/abs/1706.09697, which in particular discusses the Darboux frames. $\endgroup$
    – Ben McKay
    Commented Apr 30, 2020 at 8:26

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