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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 of such a foliation is the partition of a manifold into the orbits
of an isometric action. On one hand, recent techniques in the theory of SRF
shed light on some questions about isometric actions. On the other hand, the
theory of isometric actions feeds the other theory with examples and
motivates new questions. Furthermore, as stressed in the book of Gromoll and
Walschap, "in the last decades there has been increasing realization that
these foliations play a key role in understanding the structure of
Riemannian manifolds, particularly those with positive or nonnegative
sectional curvature". Last, but not least, the quotient space of SRF or
isometric actions provide examples of natural metric spaces with several
interesting properties.
In our research we explore some of these
connections, and discuss, among other things, quotient spaces,
generalizations of Riemannian submersions, relations between topology of the
ambient space and topological invariants of foliations. For more information
see our book or our
papers.
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