  In 2005, Bob Connelly showed that a generic framework in $\bR^d$ is globally
rigid if it has a stress matrix of maximum possible rank, and that this
sufficient condition for generic global rigidity is preserved by the
1-extension operation. His results gave a key step in the characterisation of
generic global rigidity in the plane. We extend these results to frameworks on
surfaces in $\bR^3$. For a framework on a family of concentric cylinders, cones
or ellipsoids, we show that there is a natural surface stress matrix arising
from assigning edge and vertex weights to the framework, in equilibrium at each
vertex. In the case of cylinders and ellipsoids, we show that having a maximum
rank stress matrix is sufficient to guarantee generic global rigidity on the
surface. We then show that this sufficient condition for generic global
rigidity is preserved under 1-extension and use this to make progress on the
problem of characterising generic global rigidity on the cylinder.
