  In this paper we discuss the smoothness conditions for metrics on a
cohomogeneity one manifold, i.e. metrics invariant under a Lie group whose
generic orbits are hypersurfaces. Along these hypersurfaces one describes the
metrics in terms of a collection of functions defined along a geodesic normal
to the hypersurfaces. In a neighborhood of a lower dimensional orbit the
functions must satisfy certain smoothness conditions for the metric to extend
smoothly. In general these can be quite complicated. We present a method that
makes it straightforward to compute them, and illustrate it in several
examples.
  This second version contains some improvements in exposition and a
reformulation of Theorem B, with a proof added.
