Metric Foliations in Space Forms
We have so far focused our attention mostly on the base space B of a Riemannian submersion M → B, in particular when searching for new metrics of nonnegative curvature on B. It is also interesting to look at the total space of the fibration. The very fact that there exists a Riemannian submersion from M (or more generally, that M 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 fibrations, at least in nonnegative curvature, which will be described in this chapter.
KeywordsSpace Form Normal Bundle Negative Curvature Parallel Section Riemannian Submersion
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