9 research outputs found
Riemannian foliations on contractible manifolds
We prove that Riemannian foliations on complete contractible manifolds have a closed leaf, and that all leaves are closed if one closed leaf has a finitely generated fundamental group. Under additional topological or geometric assumptions we prove that the foliation is also simple
Riemannian foliations on contractible manifolds
We prove that Riemannian foliations on complete contractible manifolds have a closed leaf, and that all leaves are closed if one closed leaf has a finitely generated fundamental group. Under additional topological or geometric assumptions we prove that the foliation is also simple
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
Singular riemannian foliations with sections, transnormal maps and basic forms
A singular riemannian foliation F on a complete riemannian manifold M is said
to admit sections if each regular point of M is contained in a complete totally
geodesic immersed submanifold (a section) that meets every leaf of F
orthogonally and whose dimension is the codimension of the regular leaves of F.
We prove that the algebra of basic forms of M relative to F is isomorphic to
the algebra of those differential forms on a section that are invariant under
the generalized Weyl pseudogroup of this section. This extends a result of
Michor for polar actions. It follows from this result that the algebra of basic
function is finitely generated if the sections are compact.
We also prove that the leaves of F coincide with the level sets of a
transnormal map (generalization of isoparametric map) if M is simply connected,
the sections are flat and the leaves of F are compact. This result extends
previous results due to Carter and West, Terng, and Heintze, Liu and Olmos.Comment: Preprint IME-USP; The final publication is available at
springerlink.com http://www.springerlink.com/content/q48682633730t831