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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Parallel focal structure and singular Riemannian foliations

Author: Dirk Töben
Journal: Trans. Amer. Math. Soc. 358 (2006), 1677-1704
MSC (2000): Primary 53C12; Secondary 53C40
Published electronically: July 26, 2005
MathSciNet review: 2186992
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Abstract: We give a necessary and sufficient condition for a submanifold with parallel focal structure to give rise to a global foliation of the ambient space by parallel and focal manifolds. We show that this is a singular Riemannian foliation with complete orthogonal transversals. For this object we construct an action on the transversals that generalizes the Weyl group action for polar actions.

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  • [A] M. Alexandrino: Singular Riemannian Foliations with Sections, Submitted article, see also math.DG/0311454.
  • [Bou] H. Boualem: Feuilletages riemanniens singuliers transversalement integrables, Composito Mathematica 95 (1995), 101-125. MR 1314698 (96e:53035)
  • [BH1] R.A. Blumenthal, J. Hebda: De Rham Decomposition Theorem for Foliated Manifolds, Ann. Inst. Fourier 33(2) (1983), 133-198. MR 0699494 (84j:53042)
  • [BH2] R.A. Blumenthal, J. Hebda: Ehresmann Connections for Foliations, Indiana Mathematical Journal 33 (1984), 597-611.MR 0749317 (85m:53024)
  • [Ew] H. Ewert: Equifocal Submanifolds in Riemannian Symmetric Spaces, Doctoral Dissertation, Universität zu Köln, 1998.
  • [HLO] E. Heintze, X. Liu, C. Olmos: Isoparametric Submanifolds and a Chevalley-type Restriction Theorem, Preprint.
  • [Kl] W. Klingenberg: Riemannian Geometry, de Gruyter, Berlin 1982. MR 0666697 (84j:53001)
  • [Mo] P. Molino: Riemannian Foliations, Progress in Mathematics Vol. 73, Birkhäuser, Boston 1988. MR 0932463 (89b:53054)
  • [Pa] R.S. Palais: A Global Formulation of the Lie Theory of Transformation Groups, Mem. Amer. Math. Soc. 22 (1957). MR 0121424 (22:12162)
  • [PaTe] R.S. Palais, C.L. Terng: Critical Point Theory and Submanifold Geometry, Geometry Lecture Notes in Math., Vol. 1353, Springer, Berlin 1988. MR 0972503 (90c:53143)
  • [PoTh] F. Podestà, G. Thorbergsson: Polar Actions on Rank One Symmetric Spaces, J. Diff. Geom. 53 (1999), no. 1, 131-175.MR 1776093 (2001j:53064)
  • [Rei] B. Reinhart: Foliated Manifolds with Bundle-like Metrics, Ann. of Math. (2) 69 (1959), 119-132. MR 0107279 (21:6004)
  • [Th1] G. Thorbergsson: Dupin Hypersurfaces, Bull. London Math. Soc. 15 (1983), 493-498. MR 0705529 (85b:53066)
  • [Th2] G. Thorbergsson: A Survey on Isoparametric Hypersurfaces and their Generalizations, Handbook of Differential Geometry, Vol. 1, Elsevier Science B.V. 2000. MR 1736861 (2001a:53097)
  • [TeTh] C.L. Terng, G. Thorbergsson: Submanifold Geometry in Symmetric Spaces, J. Diff. Geom. 42 (1995), 665-718. MR 1367405 (97k:53054)
  • [Tö] D. Töben: Submanifolds with Parallel Focal Structure, Doctoral Disseration, Universität zu Köln, 2003.

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Additional Information

Dirk Töben
Affiliation: Mathematisches Institut, Universität zu Köln, Weyerthal 86-90, 50931 Köln, Germany

Keywords: Singular Riemannian foliations, global foliation, polar actions, equifocal submanifolds, cut locus, holonomy
Received by editor(s): June 15, 2004
Published electronically: July 26, 2005
Article copyright: © Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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