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From Frenet to Cartan. The method of moving frames. (English) Zbl 1365.53001

Graduate Studies in Mathematics 178. Providence, RI: American Mathematical Society (AMS) (ISBN 978-1-4704-2952-2/hbk; 978-1-4704-3747-3/ebook). xvi, 414 p. (2017).
The present book has a high didactical and scientific quality being a very careful introduction to the method of moving frames. Besides the main theory each section contains a variety of suitable exercises to challenge the reader as well as several very interesting examples.
The content is divided into four parts. The first one, called Background material, contains the basics of differential geometry: manifolds, tensor fields, Lie groups, vector and principal bundles (in the first chapter) and differential forms (in the second chapter). Also, the last section of the second chapter is an excellent introduction to the Cartan package for Maple, used throughout the whole text. The second part, called Curves and surfaces in homogeneous spaces via the method of moving frames, covers Chapters 3–7. The ambient homogeneous geometries are: Euclidean, Minkowski, equi-affine and projective. The third part, called Applications of moving frames, contains some very active and attractive areas of research: minimal surfaces, pseudospherical surfaces and Bäcklund theorem, doubly ruled surfaces, the Cauchy-Crofton formula. The last part is Beyond the flat case: moving frames on Riemannian manifolds and deals with curves and surfaces, firstly in elliptic and hyperbolic spaces, and secondly in a general 3D Riemannian geometry.
In conclusion, this book is a very nice presentation of an essential tool of classical differential geometry. I strongly recommend it as an welcome addition to the main textbooks in geometry.

MSC:

53-02 Research exposition (monographs, survey articles) pertaining to differential geometry
53A04 Curves in Euclidean and related spaces
53C30 Differential geometry of homogeneous manifolds
58A15 Exterior differential systems (Cartan theory)
58J72 Correspondences and other transformation methods (e.g., Lie-Bäcklund) for PDEs on manifolds
53B25 Local submanifolds
53A05 Surfaces in Euclidean and related spaces
53A07 Higher-dimensional and -codimensional surfaces in Euclidean and related \(n\)-spaces

Software:

Cartan; Maple
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