Dimension of the isometry group in three-dimensional Riemannian spaces

The necessary and sufficient conditions for a three-dimensional Riemannian metric to admit a group of isometries of dimension $r$ acting on s-dimensional orbits are obtained. These conditions are Intrinsic, Deductive, Explicit and ALgorithmic and they offer an IDEAL labeling that improves previously known invariant studies.
Comment: 8 pages, 1 figure