Gauss, Peterson–Codazzi, and Ricci equations in nonholonomic frames

The isometric immersion of the $n$-dimensional pseudo-Riemannian manifold to an $m$-dimensional pseudo-Riemannian space of the constant curvature is under consideration. The manifold is assumed to be Hausdorff and orientable. Using the non-holonomic frames the author derived Gauss, Peterson–Codazzi, Ricci equations for $C^2$ immersion of this manifold into $m$-dimensional pseudo-Riemannian space of constant curvature. The main result is obtained with the use of generalized external de Rham derivation. It is found that in this context the forms of connectivity, immersion and torsion have continuous generalized exterior derivations.