Class two analogue of T.Y. Thomas's theorem and different types of embeddings of static spherically symmetric space-times

A Riemannian space of embedding class two is characterised by two symmetric tensors asub(ij), bsub(ij) and a vector ssub(i), satisfying the equations of Gauss, Codazzi and Ricci. It is proved that the Gauss equations together with one set of Codazzi equations imply the other set of Godazzi equations and the Ricci equations, provided that the matrix of the tensor bsub(ij) (or asub(ij)) is non-singular. (The class m generalisation of the result has also been suggested). The result so proved has further been utilized in finding explicitly the asub(ij)'s and bsub(ij)'s in the case of the static spherically symmetric line element. It is further indicated that the asub(ij)'s and bsub(ij)'s so obtained are responsible for the different types of embeddings of the space-time considered. (author)