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The Newton transformation and new integral formulae for foliated manifolds

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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.

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Authors and Affiliations

  1. Institute of Mathematics, Polish Academy of Sciences, ul. Śniadeckich 8, 00-956, Warsaw, Poland

    Krzysztof Andrzejewski & Paweł G. Walczak

  2. Department of Theoretical Physics II, University of Łódź, ul. Pomorska 149/153, 90-236, Lodz, Poland

    Krzysztof Andrzejewski

  3. Faculty of Mathematics and Informatics, University of Łódź, ul. Banacha 22, 90-238, Lodz, Poland

    Paweł G. Walczak

Authors
  1. Krzysztof Andrzejewski
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  2. Paweł G. Walczak
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Correspondence to Krzysztof Andrzejewski.

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Andrzejewski, K., Walczak, P.G. The Newton transformation and new integral formulae for foliated manifolds. Ann Glob Anal Geom 37, 103–111 (2010). https://doi.org/10.1007/s10455-009-9175-7

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  • DOI: https://doi.org/10.1007/s10455-009-9175-7

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