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.
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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.