Its relationship to the problem of embedding surfaces in three-dimensional Euclidean space arises from the fact that the Gauss-Codazzi equations are in this case equivalent to Cartan's equations of structure for SO(3).
The Gauss-Codazzi equations (11) for embedding [summation] can be expressed in the form of Cartan's structure equations for the group SL(2, R) as
Then, fundamental results of the theory of submanifolds can be applied and it will be seen that solving geometrically (2) amounts to solving the Gauss-Codazzi equations (40) and (41), since that would give us the curvature and torsion of a geodesic foliation of [S.sub.[gamma]].
If [M.sup.3.sub.r]([rho]) = [R.sup.3] and f [equivalent to] 1, (2) reduces to LIE (1) and it can be seen that Gauss-Codazzi equations boil down to Da Rios equations found in 1906 .