A Matrix derivation of Cartan's structural Equations

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 Expansion-Rotation


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