An approach to understanding the geometry and topology of a given Riemannian manifold $M$ is to simplify the problem by ``reducing" $M$ via its ``symmetries''. This approach becomes quite relevant when $M$ has a ``large degree'' of symmetry. An appealing feature behind this approach is that we are free to choose the meaning behind the notion of symmetry, and degree. As a notion of symmetry we consider a particular family of foliations, which admit a ``bundle like metric'', which are known as *singular Riemannian foliations*. They are a natural candidate for symmetries of a manifold, since they encompass both the notion of *group actions by isometries* and *Riemannian submersion*s, as well as solutions to PDE which are compatible with the given Riemannian metric (for example the so called *isoperimetric foliations*). This fits in the context of the so called *Grove Symmetry Program*. In this program, Grove has proposed to first consider manifolds of positive curvature with a high degree of symmetry, i.e. with a large Lie group, acting by isometries.

In the setting of the Grove Symmetry Program, and more generally in the theory of compact transformation groups, a lot of attention has been devoted to torus actions. An extension for singular Riemannian foliations is to consider foliations where the leaves are aspherical, the so called *$A$-foliations*. In this case, all the homotopy information of a leaf is contained in its fundamental group. For the particular case when $M$ is simply-connected, the leaves of highest dimension are all homeomorphic to a torus, and so these foliations resemble torus actions.

Another context in which torus actions arise is that of ``collapse'' of Riemannian manifolds. That is a procedure where we deform the geometry of the manifold to obtain, at the end, a new, possibly not smooth, metric space of lower dimension. A central object in the study of collapse with bounded curvature is that of an $F$-structure. Moreover, this is an object that generalizes the notion of local torus actions on a smooth manifold.

Having as a starting point the theories of collapse with bounded curvature and singular Riemannian foliations, the present project studies *the compatibility of a singular Riemannian foliation and the notion of collapse*. In particular we focus on $A$-foliations on simply-connceted manifolds.

## Publications

## Team Members

**Dr. Diego Corro**

Project leader

Universität zu Köln

dcorro(at)math.uni-koeln.de