394 research outputs found
Progress in the Theory of Singular Riemannian Foliations
A singular foliation is called a singular Riemannian foliation (SRF) if every
geodesic that is perpendicular to one leaf is perpendicular to every leaf it
meets. A typical example is the partition of a complete Riemannian manifold
into orbits of an isometric action.
In this survey, we provide an introduction to the theory of SRFs, leading
from the foundations to recent developments in research on this subject.
Sketches of proofs are included and useful techniques are emphasized. We study
the local structure of SRFs in general and under curvature conditions. We
review the solution of the Palais-Terng problem on integrability of the
horizontal distribution. Important special classes of SRFs, like polar and
variationally complete foliations and their connections, are treated. A
characterisation of SRFs whose leaf space is an orbifold is given. Moreover,
desingularizations of SRFs are studied and applications, e.g., to Molino's
conjecture, are presented
Polar foliations and isoparametric maps
A singular Riemannian foliation on a complete Riemannian manifold is
called a polar foliation if, for each regular point , there is an immersed
submanifold , called section, that passes through and that meets
all the leaves and always perpendicularly. A typical example of a polar
foliation is the partition of into the orbits of a polar action, i.e., an
isometric action with sections. In this work we prove that the leaves of
coincide with the level sets of a smooth map if is simply
connected. In particular, we have that the orbits of a polar action on a simply
connected space are level sets of an isoparametric map. This result extends
previous results due to the author and Gorodski, Heintze, Liu and Olmos, Carter
and West, and Terng.Comment: 9 pages; The final publication is available at springerlink.com
http://www.springerlink.com/content/c72g4q5350g513n1
Equifocality of a singular riemannian foliation
A singular foliation on a complete riemannian manifold M is said to be
riemannian if each geodesic that is perpendicular at one point to a leaf
remains perpendicular to every leaf it meets. We prove that the regular leaves
are equifocal, i.e., the end point map of a normal foliated vector field has
constant rank. This implies that we can reconstruct the singular foliation by
taking all parallel submanifolds of a regular leaf with trivial holonomy. In
addition, the end point map of a normal foliated vector field on a leaf with
trivial holonomy is a covering map. These results generalize previous results
of the authors on singular riemannian foliations with sections.Comment: 10 pages. This version contains some misprints corrections and
improvements of Corollary 1.
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