Polyhedral Space - Metric Singularities

Metric Singularities

In the study of polyhedral spaces (particularly of those that are also topological manifolds) metric singularities play a central role. Let a polyhedral space be an n-dimensional manifold. If a point in a polyhedral space that is an n-dimensional topological manifold has no neighborhood isometric to a Euclidean neighborhood in R^n, this point is said to be a metric singularity. It is a singularity of codimension k, if it has a neighborhood isometric to R^{n-k} with a metric cone. Singularities of codimension 2 are of major importance; they are characterized by a single number, the conical angle.

The singularities can also studied topologically. Then, for example, there are no topological singularities of codimension 2. In a 3-dimensional polyhedral space without a boundary (faces not glued to other faces) any point has a neighborhood homeomorphic either to an open ball or to a cone over the projective plane. In the former case, the point is necessarily a codimension 3 metric singularity. The general problem of topologically classifying singularities in polyhedral spaces is largely unresolved (apart from simple statements that e.g. any singularity is locally a cone over a spherical polyhedral space one dimension less and we can study singularities there).