  Constant curvature surfaces are constructed from the finite action solutions
of the supersymmetric $\mathbb{C}P^{N-1}$ 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.
