In the past three or four decades, there has been increasing realization that metric foliations play a key role in understanding the structure of Riemannian manifolds, particularly those with positive or nonnegative sectional curvature. In fact, all known such spaces are constructed from only a representative handful by means of metric fibrations or deformations thereof.

This text is an attempt to document some of these constructions, many of which have only appeared in journal form. The emphasis here is less on the fibration itself and more on how to use it to either construct or understand a metric with curvature of fixed sign on a given space.

From the reviews: “The book under review is one of five or six books on foliations that should be in the professional library of every geometer. … authors define the fundamental tensors of a Riemannian submersion tensors that carry over to a metric foliation on M … . gives a brief introduction to the geometry of the second tangent bundle and related topics needed for the study of metric foliations on compact space forms of non negative sectional curvature … .” (Richard H. Escobales, Jr., Mathematical Reviews, Issue 2010 h)