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**Riemannian foliations. With appendices by G. Cairns, Y. Carrière, E. Ghys, E. Salem, V. Sergiescu.
Transl. from the French by Grant Cairns.**
*(English)*
Zbl 0633.53001

Progress in Mathematics, Vol. 73. Boston-Basel: Birkhäuser. XII, 339 p.; DM 78.00 (1988).

The main text of the book splits into two parts. The first five chapters give a systematic exposition of the theory of Riemannian foliations. The aim of these chapters is to give proofs of the structure theorems for Riemannian foliations. The sixth chapter is devoted to singular Riemannian foliations.

To be more precise, the first chapter contains elements of the general theory of foliations. Chapter 2 is devoted to the transverse geometry of foliations. The author defines basic functions and forms, transverse vector fields, transverse and projectable connections on the principal bundle of transverse frames. The cohomological obstructions for the existence of a projectable connection are given. In Chapter 3, the author defines a Riemannian foliation, transverse metric, bundle-like metric, transverse Levi-Civita connection and transverse Killing field. Properties of geodesics for bundle-like metrics are studied. The author proves that if (M,\({\mathcal F})\) is a Riemannian foliation with compact leaves, then M/\({\mathcal F}\) is a Satake manifold. In the next chapter the author proves two structure theorems for transversely parallelizable and transversely complete foliations. The notions of the Atiyah sequence for a transversely complete foliation and its principal realization are introduced. A necessary and sufficient condition for the foliation to be developable is given.

Chapter 5 contains two structure theorems for Riemannian foliations. Also, the theorem of Pierrot on the structure of the singular foliation defined by the orbits of global transverse fields and the theorem of Mozgawa on the structure of Killing foliations are proved. At the end of this chapter, the author studies the cases of small codimensions. Chapter 6 is devoted to the study of singular Riemannian foliations. The reader can find a proof of a local decomposition theorem and that of the structure theorem for singular Riemannian foliations.

All chapters contain a number of exercises. The book also contains five appendices: A) “Variations on Riemannian flows” by Y. Carrière, B) “Basic cohomology and tautness of Riemannian foliations” by V. Sergiescu, C) “The duality between Riemannian foliations and geodesible foliations” by G. Cairns, D) “Riemannian foliations and pseudogroups of isometries” by E. Salem, E) “Riemannian foliations: examples and problems” by E. Ghys.

To be more precise, the first chapter contains elements of the general theory of foliations. Chapter 2 is devoted to the transverse geometry of foliations. The author defines basic functions and forms, transverse vector fields, transverse and projectable connections on the principal bundle of transverse frames. The cohomological obstructions for the existence of a projectable connection are given. In Chapter 3, the author defines a Riemannian foliation, transverse metric, bundle-like metric, transverse Levi-Civita connection and transverse Killing field. Properties of geodesics for bundle-like metrics are studied. The author proves that if (M,\({\mathcal F})\) is a Riemannian foliation with compact leaves, then M/\({\mathcal F}\) is a Satake manifold. In the next chapter the author proves two structure theorems for transversely parallelizable and transversely complete foliations. The notions of the Atiyah sequence for a transversely complete foliation and its principal realization are introduced. A necessary and sufficient condition for the foliation to be developable is given.

Chapter 5 contains two structure theorems for Riemannian foliations. Also, the theorem of Pierrot on the structure of the singular foliation defined by the orbits of global transverse fields and the theorem of Mozgawa on the structure of Killing foliations are proved. At the end of this chapter, the author studies the cases of small codimensions. Chapter 6 is devoted to the study of singular Riemannian foliations. The reader can find a proof of a local decomposition theorem and that of the structure theorem for singular Riemannian foliations.

All chapters contain a number of exercises. The book also contains five appendices: A) “Variations on Riemannian flows” by Y. Carrière, B) “Basic cohomology and tautness of Riemannian foliations” by V. Sergiescu, C) “The duality between Riemannian foliations and geodesible foliations” by G. Cairns, D) “Riemannian foliations and pseudogroups of isometries” by E. Salem, E) “Riemannian foliations: examples and problems” by E. Ghys.

Reviewer: A.Piatkowski

### MSC:

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

53C12 | Foliations (differential geometric aspects) |

57R30 | Foliations in differential topology; geometric theory |