On rigged immersions

A self-contained account is given in an efficient formalism of rigged immersions of one manifold-with-connection in another, leading to the analogues of the Gauss, Codazzi and Ricci equations discovered by Schouten. The equations expressing their interdependence are then derived and it is shown that in general one of the two sets of “Codazzi” equations is a consequence of the other set and the Gauss and Ricci equations. The formalism is specialised to the Riemannian case, where it is shown that, for large codimension (specific limits being given), all butn components of the Codazzi equations are determined by the other equations. A local theorem on the existence of rigged immersions is proved.