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2011 | 9 | 3 | 558-577
Tytuł artykułu

Integral formulae for a Riemannian manifold with two orthogonal distributions

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
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.
Wydawca
Czasopismo
Rocznik
Tom
9
Numer
3
Strony
558-577
Opis fizyczny
Daty
wydano
2011-06-01
online
2011-03-22
Twórcy
Bibliografia
  • [1] Andrzejewski K., Walczak P.G., The Newton transformation and new integral formulae for foliated manifolds, Ann. Global Anal. Geom., 2009, 37(2), 103–111 http://dx.doi.org/10.1007/s10455-009-9175-7
  • [2] Andrzejewski K., Walczak P.G., Extrinsic curvatures of distributions of arbitrary codimension, J. Geom. Phys., 2010, 60(5), 708–713 http://dx.doi.org/10.1016/j.geomphys.2010.01.003
  • [3] Asimov D., Average Gaussian curvature of leaves of foliations, Bull. Amer. Math. Soc., 1978, 84(1), 131–133 http://dx.doi.org/10.1090/S0002-9904-1978-14439-5
  • [4] Berger M., A Panoramic View of Riemannian Geometry, Springer, Berlin, 2003
  • [5] Brîz€nescu V., Slobodeanu R., Holomorphicity and the Walczak formula on Sasakian manifolds, J. Geom. Phys., 2006, 57(1), 193–207 http://dx.doi.org/10.1016/j.geomphys.2006.02.011
  • [6] Brito F., Langevin R., Rosenberg H., Intégrales de courbure sur des variétés feuilletées, J. Differential Geom., 1981, 16(1), 19–50
  • [7] Brito F.B., Naveira A.M., Total extrinsic curvature of certain distributions on closed spaces of constant curvature, Ann. Global Anal. Geom., 2000, 18(3–4), 371–383 http://dx.doi.org/10.1023/A:1006784702342
  • [8] Gray A., Pseudo-Riemannian almost product manifolds and submersions, J. Math. Mech., 1967, 16(7), 715–737
  • [9] Prudnikov A.P., Brychkov Yu.A., Marichev O.I., Integrals and Series. Vol. 3: More Special Functions, Gordon and Breach Sci. Publ., New York, 1990
  • [10] Ranjan A., Structural equations and an integral formula for foliated manifolds, Geom. Dedicata, 1986, 20(3), 85–91
  • [11] Reeb G., Sur la courbure moyenne des variétés intégrales d’une équation de Pfaff ω = 0, C. R. Math. Acad. Sci. Paris, 1950, 231, 101–102
  • [12] Rovenski V., Walczak P., Integral formulae for foliations on Riemannian manifolds, In: Differential Geometry and its Applications, Olomouc, August 27–31, 2007, World Scientific, Hackensack, 2008, 203–214
  • [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
  • [14] Rovenski V., Walczak P., Extrinsic geometric flows on foliated manifolds I, preprint available at http://arxiv.org/abs/1003.1607v2
  • [15] Rovenski V., Walczak P.G., Integral formulae on foliated symmetric spaces, Math. Ann. (in press), DOI: 10.1007/s00208-011-0637-4
  • [16] Rovenskii V., Foliations on Riemannian Manifolds and Submanifolds, Birkhäuser, Boston, 1998
  • [17] Svensson M., Holomorphic foliations, harmonic morphisms and the Walczak formula, J. Lond. Math. Soc., 2003, 68(3), 781–794 http://dx.doi.org/10.1112/S0024610703004630
  • [18] Tondeur P., Geometry of Foliations, Monogr. Math., 90, Birkhäuser, Basel, 1997
  • [19] Walczak P.G., An integral formula for a Riemannian manifold with two orthogonal complementary distributions, Colloq. Math., 1990, 58(2), 243–252
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.doi-10_2478_s11533-011-0026-y
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