# Integral formulae for a Riemannian manifold with two orthogonal distributions

Open Mathematics (2011)

• Volume: 9, Issue: 3, page 558-577
• ISSN: 2391-5455

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## Abstract

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We obtain a series of new integral formulae for a distribution of arbitrary codimension (and its orthogonal complement) given on a closed Riemannian manifold, which start from the formula by Walczak (1990) and generalize ones for foliations by several authors. For foliations on space forms our formulae reduce to the classical type formulae by Brito-Langevin-Rosenberg (1981) and Brito-Naveira (2000). The integral formulae involve the conullity tensor of a distribution, and certain components of the curvature tensor. The formulae also deal with a set of arbitrary functions depending on the scalar invariants of the co-nullity tensor. For a special choice of the functions our formulae involve the Newton transformations of the co-nullity tensor.

## How to cite

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Vladimir Rovenski. "Integral formulae for a Riemannian manifold with two orthogonal distributions." Open Mathematics 9.3 (2011): 558-577. <http://eudml.org/doc/269769>.

abstract = {We obtain a series of new integral formulae for a distribution of arbitrary codimension (and its orthogonal complement) given on a closed Riemannian manifold, which start from the formula by Walczak (1990) and generalize ones for foliations by several authors. For foliations on space forms our formulae reduce to the classical type formulae by Brito-Langevin-Rosenberg (1981) and Brito-Naveira (2000). The integral formulae involve the conullity tensor of a distribution, and certain components of the curvature tensor. The formulae also deal with a set of arbitrary functions depending on the scalar invariants of the co-nullity tensor. For a special choice of the functions our formulae involve the Newton transformations of the co-nullity tensor.},
journal = {Open Mathematics},
keywords = {Distribution; Foliation; Riemannian metric; Mean curvatures; Co-nullity tensor; Integral formula; Divergence; Newton transformations; distribution; foliation; mean curvatures; co-nullity tensor; integral formula; divergence},
language = {eng},
number = {3},
pages = {558-577},
title = {Integral formulae for a Riemannian manifold with two orthogonal distributions},
url = {http://eudml.org/doc/269769},
volume = {9},
year = {2011},
}

TY - JOUR
TI - Integral formulae for a Riemannian manifold with two orthogonal distributions
JO - Open Mathematics
PY - 2011
VL - 9
IS - 3
SP - 558
EP - 577
AB - We obtain a series of new integral formulae for a distribution of arbitrary codimension (and its orthogonal complement) given on a closed Riemannian manifold, which start from the formula by Walczak (1990) and generalize ones for foliations by several authors. For foliations on space forms our formulae reduce to the classical type formulae by Brito-Langevin-Rosenberg (1981) and Brito-Naveira (2000). The integral formulae involve the conullity tensor of a distribution, and certain components of the curvature tensor. The formulae also deal with a set of arbitrary functions depending on the scalar invariants of the co-nullity tensor. For a special choice of the functions our formulae involve the Newton transformations of the co-nullity tensor.
LA - eng
KW - Distribution; Foliation; Riemannian metric; Mean curvatures; Co-nullity tensor; Integral formula; Divergence; Newton transformations; distribution; foliation; mean curvatures; co-nullity tensor; integral formula; divergence
UR - http://eudml.org/doc/269769
ER -

## References

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13. [13] Rovenski V., Walczak P., Variational formulae for the total mean curvatures of a codimension-one distribution, In: Differential Geometry, 8th International Colloquium, Santiago de Compostela, July 7–11, 2008, World Scientific, Hackensack, 2009, 83–93 http://dx.doi.org/10.1142/9789814261173_0008
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