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Metric foliations of homogeneous three-spheres

  • Meera Mainkar
  • Benjamin SchmidtEmail author
Original Paper
First Online:
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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.

Keywords

Homogeneous spaces Naturally reductive space Metric foliations 

Mathematics Subject Classification

53C12 57R30 53C30 
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Notes

References

  1. 1.
    Brown, N., Finck, R., Spencer, M., Tapp, K., Wu, Z.: Invariant metrics with nonnegative curvature on compact Lie groups. Canad. Math. Bull. 50(1), 24–34 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Florit, L., Goertsches, O., Lytchak, A., Toeben, D.: Riemannian foliations on contractible manifolds. Muenster J. Math. 8, 1–16 (2015)MathSciNetzbMATHGoogle Scholar
  3. 3.
    Ghys, É.: Feuilletages riemanniens sur les variétés simplement connexes. Ann. Inst. Fourier (Grenoble) 34(4), 202–223 (1984)CrossRefzbMATHGoogle Scholar
  4. 4.
    Gromoll, D., Grove, K.: The low-dimensional metric foliations of Euclidean spheres. J. Differ. Geom. 28(1), 143–156 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Gromoll, D., Tapp, K.: Nonnegatively curved metrics on \(S^2 \times \mathbb{R}\). Geom. Dedicata 99, 127–136 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Gromoll, D., Walschap, G.: Metric fibrations in Euclidean space. Asian J. Math. 1(4), 716–728 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Gromoll, D., Walschap, G.: The metric fibrations of Euclidean space. J. Differ. Geom. 57(2), 233–238 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Kerin, M., Shankar, K.: Riemannian submersions from simple, compact Lie groups. Muenster J. Math. 5, 25–40 (2012)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Lee, K.B., Yi, S.: Metric foliations on hyperbolic space. J. Korean Math. Soc. 48(1), 63–82 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Lytchak, A., Wilking, B.: Riemannian foliations of spheres. Geom. Topol. 20(3), 1257–1274 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Milnor, J.: Curvatures of left invariant metrics on Lie groups. Adv. Math. 21(3), 293–329 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Munteanu, M.: One-dimensional metric foliations on compact Lie groups. Mich. Math. J. 54(1), 25–32 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Munteanu, M.: One-dimensional metric foliations on the Heisenberg Group. Proc. Am. Math. Soc. 134(6), 1791–1802 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Novikov, S.P.: The topology of foliations. (Russian). Trudy Moskov. Mat. Obsc. 14, 248–278 (1965)MathSciNetGoogle Scholar
  15. 15.
    Sekigawa, K.: On some \(3\)-dimensional curvature homogeneous spaces. Tensor (N.S.) 31(1), 87–97 (1977)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Speranca, L., Weil, S.: Metric foliations on the Euclidean space (2018). arXiv:1806.09580v1
  17. 17.
    Tricerri, F., Vanhecke, L.: homogeneous Structures on Riemannian Manifolds. London Mathematical Society Lecture Note Series, vol. 83. Cambridge University Press, Cambridge (1983)CrossRefzbMATHGoogle Scholar
  18. 18.
    Walschap, G.: Geometric vector fields on Lie groups. Differ. Geom. Appl. 7, 219–230 (1997)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Department of MathematicsCentral Michigan UniversityMount PleasantUSA
  2. 2.Department of MathematicsMichigan State UniversityEast LansingUSA

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