In information geometry, there has been extensive research on the deep connections between differential geometric structures, such as the Fisher metric and theα-connection, and the statistical theory for statistical models satisfying regularity conditions. However, the study of information geometry for non-regular statistical models is insufficient, and a one-sided truncated exponential family (oTEF) is one example of these models. In this paper, based on the asymptotic properties of maximum likelihood estimators, we provide a Riemannian metric for the oTEF. Furthermore, we demonstrate that the oTEF has anα= 1 parallel prior distribution and that the scalar curvature of a certain submodel, including the Pareto family, is a negative constant.