The Geometry of Singular Foliations

Foliation theory is a classical subject within differential geometry, since the work of Reeb in the 1950s. It studies how a given space can be decomposed smoothly into subspaces of smaller dimension, called leaves. We study decompositions in which the dimension of the leaves can vary, as for the decomposition of 3-dimensional Euclidean space into concentric spheres about the origin, together with the origin. Rather than studying the decomposition itself we study a richer structure, called Stefan-Sussmann foliation, which is known to induce automatically a group-like structure that encodes much of the geometry. In part of this project we consider Stefan-Sussmann foliations for which the ambient space is a Riemannian space or a complex space. In the former case, this means that the leaves are equidistant, as for the example mentioned above. In all these cases, our aim is to prove structure theorems for them. For instance, one aim is to show that under certain hypotheses a Stefan- Sussmann foliation - nearby a leaf - is equivalent to its linearization, which is a particularly simple kind of Stefan-Sussmann foliation. Although the emphasis is on Stefan-Sussmann foliations, we also study an extension of this notion - singular subalgebroids - which is natural from the point of view of groupoid theory.