In the usual development of Cartan's structural equations, it is presumed that
the Repere Mobile of basis vectors consists of an orthonormalized set. The
determinant of the basis frame is unity. (No expansion is allowed). The
evolution of a position vector and the basis vectors in such a space lead to
Cartan's two structural equations.
However, it is not clear that every domain admits of a global orthonormal set of
basis vectors. In fact, starting from a projective domain where the concept of
normalization is moot (vectors have lines but no scales) a different set of
structural equations can be determined, with a third structural equation related
to expansions. The method detailed in the pdf file listed below embeds the manifold of interest in a euclidean space of higher dimension (the Whitney embedding theorem). Then the manifold of interest is viewed as a subspace of the Euclidean space, using matrix partitions. The method permits a constuctive development of the Cartan structural equations, rather than presuming they exist (out ot thin air).
Also see Torsion of Translation vs
ExpansionRotation
