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

Abstract

A Riemannian space of embedding class two is characterised by two symmetric tensors a _ij , b _ij and a vector s_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 Codazzi equations and the Ricci equations, provided that the matrix of the tensor b _ij (or a _ij ) is nonsingular. (The class m generalisation of the result has also been suggested). The result so proved has further been utilized in finding explicitly the a _ij 's and b _ij 's in the case of the static spherically symmetric line element. It is further indicated that the a _ij 's and b _ij 's so obtained are responsible for the different types of embeddings of the spacetime considered.