Metric foliations of homogeneous three-spheres

Abstract

A smooth foliation of a Riemannian manifold is metric when its leaves are locally equidistant and is homogeneous when its leaves are locally orbits of a Lie group acting by isometries. Homogeneous foliations are metric foliations, but metric foliations need not be homogeneous foliations. We prove that a homogeneous three-sphere is naturally reductive if and only if all of its metric foliations are homogeneous.

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Notes

  1. 1.

    Metric foliations are also referred to as Riemannian foliations in the literature.

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Correspondence to Benjamin Schmidt.

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The authors appreciate the anonymous referee’s careful reading of a preliminary version and suggestions for improvement. The first author thanks Michigan State University for its hospitality while this research was conducted.

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Mainkar, M., Schmidt, B. Metric foliations of homogeneous three-spheres. Geom Dedicata 203, 73–84 (2019). https://doi.org/10.1007/s10711-019-00426-4

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Keywords

  • Homogeneous spaces
  • Naturally reductive space
  • Metric foliations

Mathematics Subject Classification

  • 53C12
  • 57R30
  • 53C30