44 research outputs found
Non-positive curvature and the Ptolemy inequality
We provide examples of non-locally compact geodesic Ptolemy metric spaces
which are not uniquely geodesic. On the other hand, we show that locally
compact, geodesic Ptolemy metric spaces are uniquely geodesic. Moreover, we
prove that a metric space is CAT(0) if and only if it is Busemann convex and
Ptolemy.Comment: 11 pages, 2 figure
At infinity of finite-dimensional CAT(0) spaces
We show that any filtering family of closed convex subsets of a
finite-dimensional CAT(0) space has a non-empty intersection in the visual
bordification . Using this fact, several results
known for proper CAT(0) spaces may be extended to finite-dimensional spaces,
including the existence of canonical fixed points at infinity for parabolic
isometries, algebraic and geometric restrictions on amenable group actions, and
geometric superrigidity for non-elementary actions of irreducible uniform
lattices in products of locally compact groups.Comment: An erratum filling in a gap in the proof of an application of the
main result has been included to the original pape
Nonpositive curvature and the Ptolemy inequality
We provide examples of nonlocally, compact, geodesic Ptolemy metric spaces which are not uniquely geodesic. On the other hand, we show that locally, compact, geodesic Ptolemy metric spaces are uniquely geodesic. Moreover, we prove that a metric space is CAT(0) if and only if it is Busemann convex and Ptolem
Clifford algebras and new singular Riemannian foliations in spheres
Using representations of Clifford algebras we construct indecomposable
singular Riemannian foliations on round spheres, most of which are
non-homogeneous. This generalizes the construction of non-homogeneous
isoparametric hypersurfaces due to by Ferus, Karcher and Munzner.Comment: 21 pages. Construction of foliations in the Cayley plane added.
Proofs simplified and presentation improved, according to referee's
suggestions. To appear in Geom. Funct. Ana
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
The thick-thin decomposition and the bilipschitz classification of normal surface singularities
We describe a natural decomposition of a normal complex surface singularity
into its "thick" and "thin" parts. The former is essentially metrically
conical, while the latter shrinks rapidly in thickness as it approaches the
origin. The thin part is empty if and only if the singularity is metrically
conical; the link of the singularity is then Seifert fibered. In general the
thin part will not be empty, in which case it always carries essential
topology. Our decomposition has some analogy with the Margulis thick-thin
decomposition for a negatively curved manifold. However, the geometric behavior
is very different; for example, often most of the topology of a normal surface
singularity is concentrated in the thin parts.
By refining the thick-thin decomposition, we then give a complete description
of the intrinsic bilipschitz geometry of in terms of its topology and a
finite list of numerical bilipschitz invariants.Comment: Minor corrections. To appear in Acta Mathematic
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