Metric Foliations in Space Forms

Part of the Progress in Mathematics book series (PM, volume 268)


We have so far focused our attention mostly on the base space B of a Riemannian submersion MB, 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.


Space Form Normal Bundle Negative Curvature Parallel Section Riemannian Submersion 
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© Birkhäuser Verlag AG 2009

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