The equations governing geometrical objects in ℋ space are written in terms of operators adapted to families of left shear-freeσ0= 0) cross sections of complexified null infinity (C I+). The concept of ℋ-conformai weight (HCW) is introduced, and a derivative operatorIa′, which is closely connected with the covariant derivative but which (unlike the covariant derivative) maps objects having well-defined HCW to other such objects, is defined. A function ℐ, derived from the Gaussian curvature of left shear-free slicings ofC I+ and having a well-defined HCW, is shown to contain all the curvature information for ℋ space.