Differential geometry of special mappings.

*(English)*Zbl 1337.53001
Olomouc: Palacký University, Faculty of Science (ISBN 978-80-244-4671-4/pbk). 566 p. (2015).

This is a collective work of several authors aiming to present several advances in the theory of special maps between Riemannian manifolds. The central theme are geodesic maps between various classes of Riemannian manifolds. These are diffeomorphisms between Riemannian/affine manifolds that send geodesics to geodesics. The book contains fifteen chapters written by various authors. The first five chapters are devoted to fundamentals about topological and smooth manifolds. Chapters 6–11 are devoted to an extensive analysis of geodesic maps between Riemannian manifolds with various extra structure (e.g., Einstein, Kähler, hypersurfaces in Riemannian manifolds, etc). Chapter 12 treats \(F\)-planar maps, Chapter 13 deals with holomorphically projective maps, and finally Chapters 14 and 15 are devoted to almost geodesic maps and some generalizations. The book contains interesting comments, not only of historical but also of mathematical nature (e.g., the discussion about the definition of geodesics is quite interesting).

Reviewer: Andreas Arvanitoyeorgos (Patras)

##### MSC:

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

53A99 | Classical differential geometry |

53C99 | Global differential geometry |

53C60 | Global differential geometry of Finsler spaces and generalizations (areal metrics) |

53B05 | Linear and affine connections |

53B10 | Projective connections |

53B20 | Local Riemannian geometry |

53C22 | Geodesics in global differential geometry |