By combining of the second gradient operator, the second class of integral theorems, the Gaussian-curvature-based integral theorems and the Gaussian (or spherical) mapping, a series of invariants or geometric conservation
quantities under Gaussian (or spherical) mapping are revealed. From
these mapping invariants important transformations between original
curved surface and the spherical surface are derived. The potential
applications of these invariants and transformations to geometry are
discussed.