  In this paper we study constant positive Gauss curvature $K$ surfaces in the
3-sphere $S^3$ with $0<K<1$ as well as constant negative curvature surfaces. We
show that the so-called normal Gauss map for a surface in $S^3$ with Gauss
curvature $K<1$ is Lorentz harmonic with respect to the metric induced by the
second fundamental form if and only if $K$ is constant. We give a uniform loop
group formulation for all such surfaces with $K\neq 0$, and use the generalized
d'Alembert method to construct examples. This representation gives a natural
correspondence between such surfaces with $K<0$ and those with $0<K<1$.
