define tensors

is the parametrization of the surface. Type-a index indicate the suffix such as , where .

is the curvature matrix (second derivative) of the surface at the origin.

is the third derivative at the origin.

, b=1,...,dim is the tangent vector for a'=1,...,dim-1. Here the regular type index runs 1,...,dim, whereas the type-a index runs 1,...,dim-1.

is the metric in the tangent space.