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**Introduction to Möbius differential geometry.**
*(English)*
Zbl 1040.53002

London Mathematical Society Lecture Note Series 300. Cambridge: Cambridge University Press (ISBN 0-521-53569-7/pbk). xi, 413 p. (2003).

The reviewed monograph is devoted to the Möbius differential geometry. It can be used as a consistent introduction to this field of modern geometry as well as a detailed and fundamental survey of known results. Formally, the basic content of the book consists of 8 chapters, but essentially it may be divided into three parts. In each of those parts a particular model of Möbius geometry is presented, the author introduces and analyzes in detail corresponding notions and statements. Then, as applications, various specific classes of surfaces and submanifolds are discussed whose properties become completely and clearly apparent just in terms of the model considered.

The preliminary chapter of the monograph contains fundamental notions of Riemannian geometry and differential geometry of submanifolds necessary for future presentations. Chapters 1–3 deal with the projective model of Möbius geometry, conformal flat hypersurfaces in spheres, isothermic surfaces, Guichard nets, and Willmore surfaces. In Chapters 4–5 the quaternion model and various kinds of transformations of isothermic surfaces are considered. Chapters 6–8 are devoted to the Clifford algebra model and orthogonal systems. It is noteworthy that the considered classes of surfaces and submanifolds represent good examples of particular integrable systems, they are of great interest for geometers as well as for specialists in modern soliton theory.

The author applies both, geometric methods and methods of the integrable systems theory. It allows him to give a unified presentation of the well-known classical results and new ones obtained in the 1990-s. Note that new important properties of the mentioned classical classes of surfaces were discovered when these surfaces were investigated by novel analytic methods of soliton theory.

One of the most attractive features of the book is the detailed discussion of discrete analogues of isothermic surfaces and orthogonal systems which were intensively studied in the last decade. The monograph ends with an extensive list of references (308 items) and the author’s comments on further readings for those who would like to continue studying this subject. The reviewed monograph will be of great interest for researchers specializing in differential geometry, geometric theory of integrable systems and other related fields.

The preliminary chapter of the monograph contains fundamental notions of Riemannian geometry and differential geometry of submanifolds necessary for future presentations. Chapters 1–3 deal with the projective model of Möbius geometry, conformal flat hypersurfaces in spheres, isothermic surfaces, Guichard nets, and Willmore surfaces. In Chapters 4–5 the quaternion model and various kinds of transformations of isothermic surfaces are considered. Chapters 6–8 are devoted to the Clifford algebra model and orthogonal systems. It is noteworthy that the considered classes of surfaces and submanifolds represent good examples of particular integrable systems, they are of great interest for geometers as well as for specialists in modern soliton theory.

The author applies both, geometric methods and methods of the integrable systems theory. It allows him to give a unified presentation of the well-known classical results and new ones obtained in the 1990-s. Note that new important properties of the mentioned classical classes of surfaces were discovered when these surfaces were investigated by novel analytic methods of soliton theory.

One of the most attractive features of the book is the detailed discussion of discrete analogues of isothermic surfaces and orthogonal systems which were intensively studied in the last decade. The monograph ends with an extensive list of references (308 items) and the author’s comments on further readings for those who would like to continue studying this subject. The reviewed monograph will be of great interest for researchers specializing in differential geometry, geometric theory of integrable systems and other related fields.

Reviewer: Vasyl Gorkaviy (Kharkov)

### MSC:

53-02 | Research exposition (monographs, survey articles) pertaining to differential geometry |

53A30 | Conformal differential geometry (MSC2010) |

53C40 | Global submanifolds |

53-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to differential geometry |

53A07 | Higher-dimensional and -codimensional surfaces in Euclidean and related \(n\)-spaces |

53A20 | Projective differential geometry |

53A05 | Surfaces in Euclidean and related spaces |