CRC 878 B01 - Singular metric spaces and foliations

Basic data for this project

Description

In modern differential geometry one often looks at classes of Riemannian manifolds as topological spaces. Many interesting classes of manifolds of a fixed dimension are precompact in the (pointed) Gromov Hausdorff topology. The closure of a precompact class in the class of metric spaces usually contains (singular) objects of lower dimension. Manifolds are called collapsed if they are in a sufficiently small neighborhood of such a lower dimensional object. In many cases it is expected or known that manifolds collapse along a singular metric foliation. This is one of the reasons why we study (singular) metric foliations and more specifically isometric group actions also in their own right in this project. Another connection to isometric group actions on nonnegatively curved manifolds arises since the orbit spaces of such actions are nonnegatively curved Alexandrov spaces. This observation often allows one to bound the nature of the singularities of the orbit space which in turn give structure results for the total space.