This is joint work with Chris Goddard. In 1968 Alan Weinstein
showed that any smooth closed manifold M, except for the 2-sphere, admits
a Riemannian metric so that there are no conjugate points in the cut locus
from some base point. We extend this result to show that any generic spine
for M can be made into the cut locus, again so that there are no conjugate
points. Finally, the Riemannian distance function from some base point is
of Morse type and has the property that there is one critical point for
each face F of the spine and the index is the codimension of the face.