Deformations of three-dimensional metrics

We examine three-dimensional metric deformations based on a tetrad transformation through the action the matrices of scalar fields. We describe by this approach to deformation the results obtained by Coll et al. in [1], where it is stated that any three--dimensional metric was locally obtained as a deformation of a constant curvature metric parameterized by a 2--form.To this aim, we construct the corresponding deforming matrices and provide their classification according to the properties of the scalar $\sigma$ and of the vector $\mathbf{s}$ used in [1] to deform the initial metric. The resulting causal structure of the deformed geometries is examined, too.Finally we apply our results to a spherically symmetric three geometry and to a space sector of Kerr metric.