New in the present paper is a differential-geometrical framework for analyzing statistical problems related to the Weibull distribution using a one-parameter family of affine connections. The geometry of the statistical manifold is given. A development of the relation between the J-divergence and geodesic distance is obtained. Finally, the curvatures of the affine connections for each of the curved J-space and S-space (constant curvature) on Weibull manifolds are derived.