Singular Riemannian foliations and collapse

• Applicant: Dr. Diego Corro Tapia
• Subject Area: Mathematics
• Term: from 2020 to 2023
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 441806116

The present proposal aims to study geometric deformations of a geometric space via metric foliations. This lies in the context of developing tools for the study of geometric spaces. The study of geometric spaces is one of many foundational problems in mathematics. It allows us to describe and understand the space that we live in, for example with the theory of relativity. But it also allows us to find geometric models for more abstract concepts (or structures in them), such as a collection of data points in a data base, or the parameter space of a robotic space, to name some concrete examples.An approach to describe the geometry of a given space is via its symmetries. In this direction Grove has proposed to first consider such manifolds with a large amount of symmetry, but left open the meaning of symmetry. When one considers symmetry to be a series of "rotations" in our space described by an algebraic object known as a group, this approach has proven to be successful for understanding spaces that locally look like a deformed sphere, that is that have a lower non-negative curvature bound. Recently, a more general notion of symmetry, known as a singular Riemannian foliation, has been considered for this approach. Here we assume that the geometric space can be foliated by parallel spaces. The notion of a singular Riemannian foliation is much more flexible than the one when we consider the notion given by groups. But it also produces more technical complications that we have to overcome in order to describe the geometry. Moreover, it is known that the notion of symmetries via a group differs from the notion of symmetries via a foliation, but it is not known up to what extend they agree or disagree.A very interesting phenomena in geometry is the one of collapsing a given geometric space onto an other geometric space, in a fashion that does not alter too much the curvature of the space. A concrete example is when we shrink one of the circles of a hollow donut (a torus); in this case the torus collapses to a circle. When the space is not complicated, in the sense that any loop in the given space can be continuously deformed to a point, the phenomenon of collapse implies that the space has a large amount of symmetry in the sense of groups. By determining when a foliated space does collapse, we can develop techniques to state when does the notion of symmetry via foliations agree with the notion of symmetry via groups.Even more, the phenomenon of collapsing a space is a geometric deformation. So by determining under which conditions the presence of a foliation allows us to collapse, determines new techniques for deforming a given space. This can be applied to the study of spaces with a non-negative lower curvature bound, as well in the construction of new examples of such spaces.

• DFG Programme: Priority Programmes
• Subproject of: SPP 2026: Geometry at Infinity
• International Connection: Spain, United Kingdom
• Cooperation Partner: Professor Dr. Luis Guijarro