The mechanical properties on the active surfaces of the molds or dies will be directly influenced by non-uniform distribution of the geometric intrinsic quantities such as curvature, metric invariant, and convexity-concavity of the surfaces. This paper shows an investigation made on the relationship between such geometric intrinsic quantities and the mechanical quantities. The results demonstrate that discontinuity of Gaussian curvature will lead to discontinuity of deformation and frictional dissipative energy. The frictional dissipative energy will increase with the decrease of the normal curvature on the deep-drawing path. The heat flux will increase with the increase of inflection points on the surface. These provide novel references for optimal design of the molds and dies.