Foliation by G-orbits

Computing B-Orbits on G/H

The orbits of a Borel subgroup acting on a symmetric variety G/H occur in several areas of mathematics. For example, these orbits and their closures are essential in the study of Harish Chandra modules (see [Vog83]). There are several descriptions of these orbits, but in practice it is actually very difficult and cumbersome to compute the orbits and their closures. Since the characterizations o...

Closure Relation of K Orbits on G/B

On a foliation given by the Hecke eigenform

Let F be a foliation of the modular curve, given by the vertical trajectories of the Hecke eigenform. It is shown that F is either a Strebel or a pseudo-Anosov foliation or else can be reduced to the above foliations. An application of the result is discussed.

Desingularization of Singular Riemannian Foliation

Let F be a singular Riemannian foliation on a compact Riemannian manifold M . By successive blow-ups along the strata of F we construct a regular Riemannian foliation F̂ on a compact Riemannian manifold M̂ and a desingularization map ρ̂ : M̂ → M that projects leaves of F̂ into leaves of F . This result generalizes a previous result due to Molino for the particular case of a singular Riemannian folia...

Local Index Theory over Foliation Groupoids

We give a local proof of an index theorem for a Dirac-type operator that is invariant with respect to the action of a foliation groupoid G. If M denotes the space of units of G then the input is a G-equivariant fiber bundle P → M along with a G-invariant fiberwise Dirac-type operator D on P . The index theorem is a formula for the pairing of the index of D, as an element of a certain K-theory g...