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Metric foliations of homogeneous three-spheres

  • Meera Mainkar
  • Benjamin SchmidtEmail author
Original Paper
  • 7 Downloads

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 

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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