We have so far focused our attention mostly on the base space
of a Riemannian submersion
, in particular when searching for new metrics of nonnegative curvature on
It is also interesting to look at the total space of the fibration. The very fact that there exists a Riemannian submersion from
(or more generally, that
admits a metric foliation) is a sign that the space possesses a fair amount of symmetry. One therefore expects those Riemannian manifolds with the largest amount of symmetry — namely, space forms — to be the ones that display the most variety as far as these foliations are concerned. Surprisingly, a complete classification of metric foliations on spaces of constant curvature is not yet available. There does, however, exist a classification of metric
, at least in nonnegative curvature, which will be described in this chapter.