  We study the motion of surfaces in an intrinsic formulation in which the
surface is described by its metric and curvature tensors. The evolution
equations for the six quantities contained in these tensors are reduced in
number in two cases: (i) for arbitrary surfaces, we use principal coordinates
to obtain two equations for the two principal curvatures, highlighting the
similarity with the equations of motion of a plane curve; and (ii) for surfaces
with spatially constant negative curvature, we use parameterization by
Tchebyshev nets to reduce to a single evolution equation. We also obtain
necessary and sufficient conditions for a surface to maintain spatially
constant negative curvature as it moves. One choice for the surface's normal
motion leads to the modified-Korteweg de Vries equation,the appearance of which
is explained by connections to the AKNS hierarchy and the motion of space
curves.
