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Closure of singular foliations: the proof of Molino’s conjecture

Part of: Global differential geometry Differential topology

Published online by Cambridge University Press:  13 September 2017

Marcos M. Alexandrino  and
Marco Radeschi
Marcos M. Alexandrino
Affiliation:
Universidade de São Paulo, Instituto de Matemática e Estatística, Rua do Matão 1010, 05508 090 São Paulo, Brazil email [email protected], [email protected]
Marco Radeschi
Affiliation:
University of Notre Dame, Department of Mathematics, 255 Hurley, Notre Dame, IN 46556, USA email [email protected]
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Abstract

In this paper we prove the conjecture of Molino that for every singular Riemannian foliation $(M,{\mathcal{F}})$, the partition $\overline{{\mathcal{F}}}$ given by the closures of the leaves of ${\mathcal{F}}$ is again a singular Riemannian foliation.

Keywords

singular Riemannian foliationlinearizationMolino’s conjecture

MSC classification

Primary: 53C12: Foliations (differential geometric aspects)
Secondary: 57R30: Foliations; geometric theory
Type
Research Article
Information
Compositio Mathematica , Volume 153 , Issue 12 , December 2017 , pp. 2577 - 2590
Copyright
© The Authors 2017 

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References

Alexandrino, M. M., Proofs of conjectures about singular Riemannian foliations , Geom. Dedicata 119 (2006), 219–234.Google Scholar
Alexandrino, M. M., Desingularization of singular Riemannian foliation , Geom. Dedicata 149 (2010), 397–416.Google Scholar
Alexandrino, M. M. and Lytchak, A., On smoothness of isometries between orbit spaces , in Riemannian geometry and applications – Proc. RIGA (Ed. Univ. Bucureşti, Bucharest, 2011), 17–28.Google Scholar
Alexandrino, M. M. and Radeschi, M., Smoothness of isometric flows on orbit spaces and applications to the theory of foliations , Transform. Groups 22 (2017), 1–27, doi:10.1007/s00031-016-9386-5.Google Scholar
Bolton, J., Transnormal systems , Q. J. Math. 24 (1973), 385–395.Google Scholar
Cheeger, J. and Gromoll, D., On the structure of complete manifolds of nonnegative curvature , Ann. of Math. (2) 96 (1972), 413–443.Google Scholar
Lytchak, A. and Radeschi, M., Algebraic nature of singular Riemannian foliations in spheres, J. reine angew. Math., to appear, doi:10.1515/crelle-2016-0010.Google Scholar
Mendes, R. and Radeschi, M., Smooth basic functions, Preprint (2015), arXiv:1511.06174 [math.DG].Google Scholar
Molino, P., Riemannian foliations, Progress in Mathematics, vol. 73 (Birkhäuser, Boston, MA, 1988).Google Scholar
Molino, P., Orbit-like foliations , in Geometric study of foliations, eds Mizutani, T. et al. (World Scientific, Singapore, 1994), 97–119.Google Scholar
Radeschi, M., Clifford algebras and new singular Riemannian foliations in spheres , Geom. Funct. Anal. 24 (2014), 1660–1682.Google Scholar
Sussmann, H. J., Orbits of families of vector fields and integrability of distributions , Trans. Amer. Math. Soc. 180 (1973), 171–188.Google Scholar
Toeben, D., Parallel focal structure and singular Riemannian foliations , Trans. Amer. Math. Soc. 358 (2006), 1677–1704.Google Scholar
Wilking, B., A duality theorem for Riemannian foliations in nonnegative sectional curvature , Geom. Funct. Anal. 17 (2007), 1297–1320.Google Scholar