The classical relation between two-dimensional spaces of constant curvature and certain nonlinear partial differential equations is formulated in group-theoretic terms by means of the underlying semisimple isometry group. Rather than working with the metric and curvature in the given constant curvature space, it is then possible to consider the equivalent system consisting of a pair of first-order partial differential equations in flat space for two so(2,1) vectors satisfying a pair of SO(2,1)-invariant algebraic constraints. Such equations determine a SL(2,R) principal bundle with flat connection. The construction is carried out in detail for the case of the sine-Gordon equation. The connection is shown to give rise to a spectral problem of the inverse scattering type by means of a gauge transformation. Bäcklund transformations are shown to be automorphisms of the connection characterized by a certain so(2,1) null vector. The generation of solutions of the nonlinear equation from known ones is seen to be determined by gauge transformations leaving invariant a fixed set of gauge conditions.