Three-dimensional metrics as deformations of a constant curvature metric

Any three-dimensional metric g may be locally obtained from a constant curvature metric, h, by a deformation like g = sigmah + epsilon s x s, where sigma and s are respectively a scalar and a one-form, the sign epsilon = +/-1 and a functional relation between sigma and the Riemannian norm of s can be arbitrarily prescribed. The general interest of this result in geometry and physics, and the related open problems, are stressed.