Constant curvature surfaces are constructed from the finite action solutions of the supersymmetric CPN-1 sigma model. It is shown that there is a unique holomorphic solution which leads to constant curvature surfaces: the generalized Veronese curve. We give a general criterion to construct non-holomorphic solutions of the model. We extend our analysis to general supersymmetric Grassmannian models. (C) 2015 AIP Publishing LLC.