  A result due in its various parts to Hendrickson, Connelly, and Jackson and
Jord\'an, provides a purely combinatorial characterisation of global rigidity
for generic bar-joint frameworks in $\mathbb{R}^2$. The analogous conditions
are known to be insufficient to characterise generic global rigidity in higher
dimensions. Recently Laman-type characterisations of rigidity have been
obtained for generic frameworks in $\mathbb{R}^3$ when the vertices are
constrained to lie on various surfaces, such as the cylinder and the cone. In
this paper we obtain analogues of Hendrickson's necessary conditions for the
global rigidity of generic frameworks on the cylinder, cone and ellipsoid.
