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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
Posted:
July 26, 2005
MathSciNet review:
2186992
Full-text PDF Free Access
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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.
References
- [A]
M. Alexandrino: Singular Riemannian Foliations with Sections, Submitted article, see also http://arxiv.org 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
Email:
dtoeben@math.uni-koeln.de
DOI:
http://dx.doi.org/10.1090/S0002-9947-05-03829-8
PII:
S 0002-9947(05)03829-8
Keywords:
Singular Riemannian foliations,
global foliation,
polar actions,
equifocal submanifolds,
cut locus,
holonomy
Received by editor(s):
June 15, 2004
Posted:
July 26, 2005
Article copyright:
© Copyright 2005 American Mathematical Society
The copyright for this article reverts to public domain after
28 years from publication.
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