Three-Dimensional Metrics as Deformations of a Constant Curvature Metric

摘要

Any three-dimensional metric g may be locally obtained from a constant curvature metric, h, by a deformation like $$g = sigma h + in s otimes s,$$ where σ and s are respectively a scalar and a one-form, the sign ∈ = ±1 and a functional relation between σ 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.