ISOMETRY GROUPS OF 3-DIMENSIONAL RIEMANNIAN METRICS

The necessary and sufficient conditions for a three-dimensional Riemannian metric to admit a group G(r) of isometries acting on s-dimensional orbits are given. This provides the list of (abstract) groups that can act isometrically and maximally on such metrics. The conditions are expressed in terms of the eigenvalues and eigenvectors of the Ricci tensor. In any case, the order of differentiability of these data necessary to determine the isometry group is less than 4.