Hyperpolar homogeneous foliations on symmetric spaces of noncompact type

Berndt,Jürgen King’s College London Díaz-Ramos University of Santiago de Compostela Hiroshi Tamaru Hiroshima University

Differential Geometry mathscidoc:1609.10197

Journal of Differential Geometry, 86, (2), 191-235, 2010
A foliation F on a Riemannian manifold M is hyperpolar if it admits a flat section, that is, a connected closed flat submanifold of M that intersects each leaf of F orthogonally. In this article we classify the hyperpolar homogeneous foliations on every Riemannian symmetric space M of noncompact type. These foliations are constructed as follows. Let  be an orthogonal subset of a set of simple roots associated with the symmetric space M. Then  determines a horospherical decomposition M = Fs ×ErankM−||×N, where Fs  is the Riemannian product of || symmetric spaces of rank one. Every hyperpolar homogeneous foliation on M is isometrically congruent to the product of the following objects: a particular homogeneous codimension one foliation on each symmetric space of rank one in Fs , a foliation by parallel affine subspaces on the Euclidean space ErankM−||, and the horocycle subgroup N of the parabolic subgroup of the isometry group of M determined by .
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@inproceedings{berndt,jürgen2010hyperpolar,
  title={HYPERPOLAR HOMOGENEOUS FOLIATIONS ON SYMMETRIC SPACES OF NONCOMPACT TYPE},
  author={Berndt,Jürgen, Díaz-Ramos, and Hiroshi Tamaru},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160911224849325819854},
  booktitle={Journal of Differential Geometry},
  volume={86},
  number={2},
  pages={191-235},
  year={2010},
}
Berndt,Jürgen, Díaz-Ramos, and Hiroshi Tamaru. HYPERPOLAR HOMOGENEOUS FOLIATIONS ON SYMMETRIC SPACES OF NONCOMPACT TYPE. 2010. Vol. 86. In Journal of Differential Geometry. pp.191-235. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20160911224849325819854.
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