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**Foliations on Riemannian manifolds.**
*(English)*
Zbl 0643.53024

Universitext. New York etc.: Springer-Verlag. ix, 247 p. DM 58.00 (1988).

This book is devoted to some aspects of Riemannian geometry of foliations of Riemannian manifolds. The second fundamental tensors of leaves of a foliation \({\mathcal F}\) glue together to a (1,1)-tensor field A on the foliated Riemannian manifold M. The invariants of A, especially its trace \(\kappa\), are the main objects of interest here.

Sections 1-4 contain a brief introduction into the general theory of foliations. Section 5 is devoted to totally geodesic foliations \((A=0)\) and their relation with Riemannian foliations and bundle-like metrics in the sense of B. Reinhart [Ann. Math., II. Ser. 69, 119-132 (1959; Zbl 0122.166)]. Section 6 contains some general results and formulae concerning A and \(\kappa\). Also, several results about so-called harmonic foliations \((\kappa =0)\) are included. These results are applied in Sections 7 and 8 to codimension-one foliations. In Section 9, basic forms and the basic cohomology of a foliation are defined. Also, infinitesimal automorphisms of a foliated space are considered here.

Section 10 contains applications of the results of the preceding sections to non-singular flows considered as one-dimensional foliations. In Section 11, Lie foliations are introduced to get some result about their basic cohomology. This result is applied in Section 12 to prove some duality theorems for the basic cohomology of Riemannian foliations. Finally, Section 13 contains a brief information about a recent result by the author, F. Kamber and E. Ruh [to appear in J. Differ. Geom. 27 (see the preview in Zbl 0613.53013)] who compare Riemannian and transversely symmetric foliations. The text is accompanied with a large bibliography on foliations (more than 1000 publications).

Sections 1-4 contain a brief introduction into the general theory of foliations. Section 5 is devoted to totally geodesic foliations \((A=0)\) and their relation with Riemannian foliations and bundle-like metrics in the sense of B. Reinhart [Ann. Math., II. Ser. 69, 119-132 (1959; Zbl 0122.166)]. Section 6 contains some general results and formulae concerning A and \(\kappa\). Also, several results about so-called harmonic foliations \((\kappa =0)\) are included. These results are applied in Sections 7 and 8 to codimension-one foliations. In Section 9, basic forms and the basic cohomology of a foliation are defined. Also, infinitesimal automorphisms of a foliated space are considered here.

Section 10 contains applications of the results of the preceding sections to non-singular flows considered as one-dimensional foliations. In Section 11, Lie foliations are introduced to get some result about their basic cohomology. This result is applied in Section 12 to prove some duality theorems for the basic cohomology of Riemannian foliations. Finally, Section 13 contains a brief information about a recent result by the author, F. Kamber and E. Ruh [to appear in J. Differ. Geom. 27 (see the preview in Zbl 0613.53013)] who compare Riemannian and transversely symmetric foliations. The text is accompanied with a large bibliography on foliations (more than 1000 publications).

Reviewer: P.Walczak

### MSC:

53C12 | Foliations (differential geometric aspects) |

57R30 | Foliations in differential topology; geometric theory |

37C85 | Dynamics induced by group actions other than \(\mathbb{Z}\) and \(\mathbb{R}\), and \(\mathbb{C}\) |

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