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**Geodesic mappings of manifolds with affine connection.**
*(English)*
Zbl 1176.53004

Olomouc: Palacký University, Faculty of Science (ISBN 978-80-244-2168-1/pbk). 220 p. (2008).

This book is devoted to geodesic mappings of manifolds with affine (linear) connection and (pseudo-) Riemannian spaces. A geodesic mapping here is a diffeomorphism of one n-dimensional differentiable manifold with linear connection onto another such that geodesics are mapped onto geodesics; the affine parameter is not preserved in general. Methods of tensor analysis and geometric objects are used mostly locally, in the classical manner. Historical remarks are included as footnotes. A particular attention is devoted to existence of geodesic mappings in case when the target manifold, or both the source and the target manifolds, are of special types, e.g. Riemannian spaces, Einstein, Kähler, Weyl spaces or semisymmetric spaces.

The first two introductory chapters of the monograph are devoted to a summary of basic concepts of the manifolds with affine connection and Riemannian spaces: the theory of surfaces and the theory of manifolds with a geometric structure. In 2.5 the theory of partial differential equations of Cauchy type (in Euclidean space as well as in manifolds) is explained. Applications are demonstrated by a survey of papers in which existence theorems on geodesic, almost geodesic, holomorphically projective mappings (with particular source or target spaces), conformal mappings of Riemannian spaces onto Einstein spaces, infinitesimal geodesic deformations etc. are reduced just to the decision of a Cauchy-type system of PDEs.

The concept of geodesic and the role of the canonical parameter are discussed thoroughly in chapter three, and new insight in variational properties of geodesics is given in 3.4. Chapter 4 recalls some classical results on geodesic mappings some of them being known since the times of T. Levi-Civita. In chapters 4–10 results of J. Mikeš and his co-workers are stated, some quite recent, some contemporary and fresh ones. They are devoted mostly to geodesic mappings and deformations of special types of manifolds. Let us mention two new and a bit surprising results: A manifold with affine connection is locally projectively equivalent to an affine manifold (Theorem 4.5). If an Einstein space admits a non-trivial geodesic mapping onto a Riemannian space then the target space is also an Einstein space (Theorem 7.6).

Theorem 5.11 characterizes the situation of existence of a geodesic mapping of a Riemannian manifold V of non-constant curvature onto a semisymmetric equiaffine manifold: V must be an equidistant manifold and its curvature tensor has special properties. In Chapter 10, the above technics are used in a different topic: Abstract elementary geometries are studied from the view-point of existence of a differentiable structure and affine connection for which given abstract lines can be just geodesics to the connection.

In commentaries and remarks, some incorrect results published by other authors are corrected. The bibliography consists of 425 titles. The subject index and the name index are included.

The book is warmly recommended to specialist in mathematics, physicists and especially to PhD students interested in the topic.

The first two introductory chapters of the monograph are devoted to a summary of basic concepts of the manifolds with affine connection and Riemannian spaces: the theory of surfaces and the theory of manifolds with a geometric structure. In 2.5 the theory of partial differential equations of Cauchy type (in Euclidean space as well as in manifolds) is explained. Applications are demonstrated by a survey of papers in which existence theorems on geodesic, almost geodesic, holomorphically projective mappings (with particular source or target spaces), conformal mappings of Riemannian spaces onto Einstein spaces, infinitesimal geodesic deformations etc. are reduced just to the decision of a Cauchy-type system of PDEs.

The concept of geodesic and the role of the canonical parameter are discussed thoroughly in chapter three, and new insight in variational properties of geodesics is given in 3.4. Chapter 4 recalls some classical results on geodesic mappings some of them being known since the times of T. Levi-Civita. In chapters 4–10 results of J. Mikeš and his co-workers are stated, some quite recent, some contemporary and fresh ones. They are devoted mostly to geodesic mappings and deformations of special types of manifolds. Let us mention two new and a bit surprising results: A manifold with affine connection is locally projectively equivalent to an affine manifold (Theorem 4.5). If an Einstein space admits a non-trivial geodesic mapping onto a Riemannian space then the target space is also an Einstein space (Theorem 7.6).

Theorem 5.11 characterizes the situation of existence of a geodesic mapping of a Riemannian manifold V of non-constant curvature onto a semisymmetric equiaffine manifold: V must be an equidistant manifold and its curvature tensor has special properties. In Chapter 10, the above technics are used in a different topic: Abstract elementary geometries are studied from the view-point of existence of a differentiable structure and affine connection for which given abstract lines can be just geodesics to the connection.

In commentaries and remarks, some incorrect results published by other authors are corrected. The bibliography consists of 425 titles. The subject index and the name index are included.

The book is warmly recommended to specialist in mathematics, physicists and especially to PhD students interested in the topic.

Reviewer: Jan Kurek (Lublin)

### MSC:

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

53B05 | Linear and affine connections |

53B20 | Local Riemannian geometry |

53B21 | Methods of local Riemannian geometry |

53B25 | Local submanifolds |

53B30 | Local differential geometry of Lorentz metrics, indefinite metrics |

53B35 | Local differential geometry of Hermitian and Kählerian structures |

53A45 | Differential geometric aspects in vector and tensor analysis |

53C22 | Geodesics in global differential geometry |

53C24 | Rigidity results |

53C25 | Special Riemannian manifolds (Einstein, Sasakian, etc.) |

53C50 | Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics |

51H25 | Geometries with differentiable structure |

58B20 | Riemannian, Finsler and other geometric structures on infinite-dimensional manifolds |

58J32 | Boundary value problems on manifolds |

35F99 | General first-order partial differential equations and systems of first-order partial differential equations |

35N99 | Overdetermined problems for partial differential equations and systems of partial differential equations |

35R99 | Miscellaneous topics in partial differential equations |