The role of total and Gaussian curvature in the study of convex surfaces

Busemann and Feller proved a remarkable theorem: Every convex surface is twice differentiable almost everywhere. The Gaussian curvature of a surface is defined as the limit of the ratio of the area of the spherical map to the area of the domain. The total curvature is equal to the product of the principal normal curvatures. There may be points of the surface where the Gaussian curvature exists, but the total curvature does not exist. Since at these points no normal section has a definite curvature.