Structure of equicontinuous foliated spaces

Resumen

Structure of equicontinuous foliated spaces A foliation is called Riemannian when it is transversely rigid in the sense that there is a Riemannian metric on transversals that is invariant by holonomy transformations (sliding transversals along the leaves). For these foliations, a theory due to P. Molino describes its structure, reducing much of their study first to the particular case of transversely parallelizable (TP) foliations, and finally to the case of Lie foliations. On the other hand, there is a generalization of the concept of foliation, called foliated space, which is kind of a foliation of an arbitrary topological space by leaves that still are manifolds. They are relevant in different fields, such as Arimethics, or even in Folition Theory because minimal sets are foliated spaces. In this area, the role of Riemannian foliations is played by equicontinuos foliated spaces, the role of the TP foliations is played by homogeneous foliated spaces, and the role of Lie foliations is played by the foliated spaces whose transverse dynamics is modeled by the local action of a local group on itself by local left translations. The first goal of the thesis is to develop a theory similar to that of P. Molino, describing equicontinuos foliated spaces by using these more rigid types of foliated spaces. We will also look for examples and applications of this theory; in particular, we will try to use it to generalize invariants of Riemannian foliation theory.