The Newton transformation and new integral formulae for foliated manifolds |
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Authors: | Krzysztof Andrzejewski Paweł G Walczak |
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Institution: | 1.Institute of Mathematics,Polish Academy of Sciences,Warsaw,Poland;2.Department of Theoretical Physics II,University of ?ód?,Lodz,Poland;3.Faculty of Mathematics and Informatics,University of ?ód?,Lodz,Poland |
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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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