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


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.


Foliations Newton transformation Shape operator rth mean curvature 

Mathematics Subject Classification (2000)

53C12 53A10 53C42 


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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

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