A Riemannian cohomogeneity one manifold is a manifold M acted on isometrically by a Lie group G with codimension one principal orbits. It is easy to describe the set of the G-invariant Riemannian metrics in the regular part of M but, when there are singular orbits, not all of these metrics extend smoothly to the singular set.

We discuss a method to determine the conditions that guarantee a smooth extension to the singular set and present examples of how this can be used to construct examples of Einstein metrics in a tubular neighborhood of a singular orbit and to find obstructions to the existence of nonnegatively curved metrics on compact cohomogeneity one 7 manifolds. (Joint with W. Ziller)