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Winter 2004 Singular Riemannian foliations with sections
Marcos M. Alexandrino
Illinois J. Math. 48(4): 1163-1182 (Winter 2004). DOI: 10.1215/ijm/1258138504

Abstract

A singular foliation on a complete riemannian manifold is said to be riemannian if every geodesic that is perpendicular at one point to a leaf remains perpendicular to every leaf it meets. In this paper, we study singular riemannian foliations with sections. A section is a totally geodesic complete immersed submanifold that meets each leaf orthogonally and whose dimension is the codimension of the regular leaves.

We prove here that the restriction of the foliation to a slice of a leaf is diffeomorphic to an isoparametric foliation on an open set of an euclidean space. This result provides local information about the singular foliation and in particular about the singular stratification of the foliation. It allows us to describe the plaques of the foliation as level sets of a transnormal map (a generalization of an isoparametric map). We also prove that the regular leaves of a singular riemannian foliation with sections are locally equifocal. We use this property to define a singular holonomy. Then we establish some results about this singular holonomy and illustrate them with a couple of examples.

Citation

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Marcos M. Alexandrino. "Singular Riemannian foliations with sections." Illinois J. Math. 48 (4) 1163 - 1182, Winter 2004. https://doi.org/10.1215/ijm/1258138504

Information

Published: Winter 2004
First available in Project Euclid: 13 November 2009

zbMATH: 1069.53030
MathSciNet: MR2113670
Digital Object Identifier: 10.1215/ijm/1258138504

Subjects:
Primary: 53C12
Secondary: 57R30

Rights: Copyright © 2004 University of Illinois at Urbana-Champaign

Vol.48 • No. 4 • Winter 2004
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