Ricci Collineations of the Bianchi Types I and III, and Kantowski-Sachs Spacetimes

Ricci collineations of the Bianchi types I and III, and Kantowski-Sachs space- times are classified according to their Ricci collineation vector (RCV) field of the form (i)-(iv) one component of $\xi^a (x^b)$ is nonzero, (v)-(x) two components of $\xi^a (x^b)$ are nonzero, and (xi)-(xiv) three components of $\xi^a (x^b)$ are nonzero. Their relation with isometries of the space-times is established. In case (v), when $det(R_{ab}) = 0$, some metrics are found under the time transformation, in which some of these metrics are known, and the other ones new. Finally, the family of contracted Ricci collineations (CRC) are presented.