The generalized method of moments (GMM) attains the semiparametric efficiency bound when the optimal weight matrix is chosen. In this study, we characterize the efficient choice of the weight matrix from the viewpoint of differential geometry. We induce a metric for the GMM manifold from a linear space in which the model set is embedded. Simultaneously, using the asymptotic normality of the GMM estimators, we define another metric of the manifold. In conclusion, we prove that the two metrics coincide when an optimal weight matrix is employed.