Metric foliations and curvature / Detlef Gromoll, Gerard Walschap

The book contains the following chapters:
I.

Submersions, Foliations and Metrics,
II.

Basic Constructions and Examples,
III.

Open Manifolds of Nonnegative Curvature,
IV.

Metric Foliations in Space Forms.

In Chapter I, the authors introduce the main concepts and tools which are used in the book. Relations between curvature of the ambient space and of the base are considered. Also connections between geodesics and Jacobi fields of both manifolds are studied. At the end of this chapter, the authors describe the Wilking’s dual foliation.

In Chapter II, the authors study the influence of warping of fibres on the curvature of the ambient space. This is used to introduce warped products. Then the class of Riemannian submersions generated by isometric group actions is introduced. The rest of the chapter is devoted to the construction of spaces of positive or nonnegative curvature by means of submersions. Fundamental examples are given.

The structure of complete, noncompact manifolds with nonnegative curvature is studied in Chapter III (including Cheeger-Gromoll soul construction). The authors present Perelman’s proof of the generalized soul theorem, and Wilking’s work on smoothness of the metric projection onto the soul. The converse of the soul theorem is also discussed.

Chapter IV deals with the problems of classifying foliations on spaces of constant curvature. (Zentralblatt)