Advertisement

Annals of Global Analysis and Geometry

, Volume 37, Issue 2, pp 103–111 | Cite as

The Newton transformation and new integral formulae for foliated manifolds

  • Krzysztof AndrzejewskiEmail author
  • Paweł G. Walczak
Original Paper

Abstract

In this article, we show that the Newton transformations of the shape operator can be applied successfully to foliated manifolds. Using these transformations, we generalize known integral formulae (due to Brito–Langevin–Rosenberg, Ranjan, Walczak, etc.) for foliations of codimension one. We obtain integral formulae involving rth mean curvature of the second fundamental form of a foliation, the Jacobi operator in the direction orthogonal to the foliation, and their products. We apply our formulae to totally umbilical foliations and foliations whose leaves have constant second order mean curvature.

Keywords

Foliations Newton transformation Shape operator rth mean curvature 

Mathematics Subject Classification (2000)

53C12 53A10 53C42 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Alías L.J., Colares A.G.: Uniqueness of spacelike hypersurfaces with constant higher order mean curvature in generalized Robertson-Walker spacetimes. Math. Proc. Camb. Phil. Soc. 143, 703–729 (2007)zbMATHCrossRefGoogle Scholar
  2. 2.
    Alías L.J., Malacarne J.M.: Constant scalar curvature hypersurfaces with spherical boundary in Euclidean space. Rev. Mat. Iberoamericana 18, 431–432 (2002)zbMATHMathSciNetGoogle Scholar
  3. 3.
    Alías L.J., de Lira S., Malacarne J.M.: Constant higher-order mean curvature hypersurfaces in Riemannian spaces. J Inst. of Math. Jussieu 5(4), 527–562 (2006)zbMATHCrossRefGoogle Scholar
  4. 4.
    Asimov D.: Average Gaussian curvature of leaves of foliations. Bull. Amer. Math. Soc. 84, 131–133 (1978)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Baird, P., Wood J.C.: Harmonic morphisms between Riemannian manifolds. London Mathematical Society Monograph (N.S.) No. 29. Oxford University Press, Oxford (2003)Google Scholar
  6. 6.
    Barbosa J.L.M., Colares A.G.: Stability of hypersurfaces with constant r-mean curvature. Ann. Global Anal. Geom. 15, 277–297 (1997)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Barbosa J.L.M., Kenmotsu K., Oshikiri G.: Foliations by hypersurfaces with constant mean curvature. Mat. Z. 207, 97–108 (1991)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Barros A., Sousa P.: Compact graphs over a sphere of constant second order mean curvature. Proc. Am. Math. Soc. 137(9), 3105–3114 (2009)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Brinzanescu V., Slobodeanu R.: Holomorphicity and Walczak formula on Sasakian manifolds. J. Geom. Phys. 57, 193–207 (2006)CrossRefMathSciNetGoogle Scholar
  10. 10.
    Brito F., Naveira A.M.: Total extrinsic curvature of certain distributions on closed spaces of constant curvature. Ann. Global Anal. Geom. 18, 371–383 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Brito F., Langevin R., Rosenberg H.: Intégrales de courbure sur des variétés feuilletées. J. Diff. Geom. 16, 19–50 (1981)zbMATHMathSciNetGoogle Scholar
  12. 12.
    Cheng X., Rosenberg H.: Embedded positive constant r-mean curvature hypersurfaces in M m × R. An. Acad. Brasil. CiÊn 77(2), 183–199 (2005)zbMATHMathSciNetGoogle Scholar
  13. 13.
    Czarnecki M., Walczak P.: Extrinsic geometry of foliations. In: Walczak (eds) Foliations 2005. World Scientific, Singapore, pp. 149–167 (2006)CrossRefGoogle Scholar
  14. 14.
    Ranjan A.: Structural equations and an integral formula for foliated manifolds. Geom. Dedicata 20, 85–91 (1986)zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Ros A.: Compact hypersurfaces with constant higher order mean curvatures. Rev. Mat. Iberoamericana 3, 447–453 (1987)zbMATHMathSciNetGoogle Scholar
  16. 16.
    Rosenberg H.: Hypersurfaces of constant curvature in space forms. Bull. Sci. Math. 117, 211–239 (1993)zbMATHMathSciNetGoogle Scholar
  17. 17.
    Rovenski, V., Walczak, P.: Integral formulae on foliated symmetric spaces. Preprint (2007)Google Scholar
  18. 18.
    Rovenski, V., Walczak, P.: Integral formulae for foliations on Riemannian manifolds. In: Differential Geometry and its Application, Proceedings of the 10th International Conference on DGA 2007, Olomouc, pp. 203–214. World Scientific, Singapore (2008)Google Scholar
  19. 19.
    Rummler H.: Quelques notions simples en Géométrie Riemanniene et leurs applications sur feuilletages compacts. Comment Math. Helv. 54, 224–239 (1979)zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Svensson M.: Holomorphic foliations, harmonic morphisms and the Walczak formula. J. London Math. Soc. 68, 781–794 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Tondeur P.: Geometry of foliations. Birkhauser Verlag, Basel (1997)zbMATHGoogle Scholar
  22. 22.
    Walczak P.: An integral formula for a Riemannian manifold with two orthogonal complementary distributions. Colloq. Math. 58, 243–252 (1990)zbMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2009

Authors and Affiliations

  1. 1.Institute of MathematicsPolish Academy of SciencesWarsawPoland
  2. 2.Department of Theoretical Physics IIUniversity of ŁódźLodzPoland
  3. 3.Faculty of Mathematics and InformaticsUniversity of ŁódźLodzPoland

Personalised recommendations