New integral formulae for two complementary orthogonal distributions on Riemannian manifolds

详细信息    查看全文

• 作者：Magdalena Lu?yńczyk ; Pawe? Walczak
• 关键词：Riemannian manifold ; Distribution ; Foliation ; Integral formula ; Primary 53C15 ; Secondary 53C12
• 刊名：Annals of Global Analysis and Geometry
• 出版年：2015
• 出版时间：August 2015
• 年：2015
• 卷：48
• 期：2
• 页码：195-209
• 全文大小：452 KB
• 参考文献：1.Alexander, J.W.: Note on Riemann spaces. Bull. Am. Math. Soc. 26, 370-72 (1920)View Article
  2.Andrzejewski, K.: The Newton Transformation and Extrinsic Curvatures of Foliations and Distributions, Thesis. IMPAN, Warsaw (2010)
  3.Andrzejewski, K., Walczak, P.: The Newton transformation and new integral formulae for foliated manifolds. Ann. Global Anal. Geom. 37(2), 103-11 (2010)MathSciNet View Article
  4.Asimow, D.: Average Gaussian curvature of leaves of foliations. Bull. Am. Math. Soc. 84, 131-33 (1978)View Article
  5.Baird, P., Wood, J.C.: Harmonic morphisms between Riemannian manifolds, London Math. Society Monographs (N.S.), vol. 29. Oxford University Press, Oxford (2003)
  6.Brito, F., Langevin, R., Rosenberg, H.: Intégrales de courbure sur des variétés feuilletées. J. Differ. Geom. 16, 19-0 (1981)MathSciNet
  7.Brito, F., Naveira, A.M.: Total extrinsic curvature of certain distributions on closed spaces of constant curvature. Ann. Global Anal. Geom 18, 371-83 (2000)MathSciNet View Article
  8.Brito, F., Walczak, P.: On the energy of unit vector fields with isolated singularities. Ann. Pol. Math. 73, 269-73 (2000)MathSciNet
  9.Langevin, R., Walczak, P.: Conformal geometry of foliations. Geom. Dedicata 132, 135-78 (2008)MathSciNet View Article
  10.Lawson, T.: Open book decompositions for odd dimensional manifolds. Topology 17, 189-92 (1978)MathSciNet View Article
  11.Ranjan, A.: Structural equations and an integral formula for foliated manifolds. Geom. Dedicata 20, 85-1 (1986)MathSciNet View Article
  12.Reeb, G.: Sur la courboure moyenne des variétés intégrales d’une équation de Pfaff \(\omega = 0\) . C. R. Acad. Sci. Paris 231, 101-02 (1950)MathSciNet
  13.Reinhart, B.L.: Foliated manifolds with bundle-like metrics. Ann. Math. 69, 119-32 (1959)MathSciNet View Article
  14.Rovenski, V.: The partial Ricci flow for foliations. In: Rovenski, V., Walczak, P. (eds.) Geometry and Its Applications, pp. 125-55. Springer, Berlin (2014)View Article
  15.Rovenski, V., Walczak, P.: Integral formulae for foliated symmetric spaces. Math. Ann. 352, 223-37 (2012)MathSciNet View Article
  16.Rovenski, V., Walczak, P.: Topics in Extrinsic Geometry of Codimension-One Foliations. Springer Briefs in Mathematics. Springer, Berlin (2011)View Article
  17.Rovenski, V., Wolak, R.: Deforming metrics of foliations. Central Eur. J. Math. 11, 1039-055 (2013)MathSciNet
  18.Svensson, M.: Holomorphic foliations, harmonic morphisms and the Walczak formula. J. Lond. Math. Soc. 68, 781-94 (2003)MathSciNet View Article
  19.Walczak, P.: An integral formula for Riemannian manifold with two orthogonal complementary distributions. Coll. Math. 58, 243-52 (1990)MathSciNet
  20.Winkelkemper, H.E.: Manifolds as open books. Bull. Am. Math. Soc. 79, 45-1 (1973)View Article
  
• 作者单位：Magdalena Lu?yńczyk (1)
  Pawe? Walczak (1)
  
  1. Wydzia? Matematyki i Informatyki, Uniwersytet ?ódzki, ?ód?, Poland
  
• 刊物类别：Mathematics and Statistics
• 刊物主题：Mathematics
  Analysis
  Statistics for Business, Economics, Mathematical Finance and Insurance
  Mathematical and Computational Physics
  Group Theory and Generalizations
  Geometry
  
• 出版者：Springer Netherlands
• ISSN：1572-9060

文摘

We derive and apply a new integral formula for a closed Riemannian manifold equipped with a pair of complementary orthogonal distributions (plane fields). The integrand depends on the second fundamental forms and integrability tensors of the distributions, their covariant derivatives, and of the Ricci curvature of the ambient manifold. Also, we discuss some applications of this formula and of another formula of this sort, the one obtained earlier by the second author, and show that both formulae may hold when the distributions are defined only outside a “reasonable-closed subset of the manifold under consideration.