  which preserves the metric up to a positive and smooth but otherwise  arbitrary scalar factor or gauge function  \(\lambda(x(p))\). In the case of a pseudo-Riemannian  structure such a gauge transformation leaves the light cones  unaltered. The angle between two vectors at \(p\) is given by  (11). Clearly, the gauge transformation  \(\overline{g}_{ij}(x(p)) = \lambda(x(p))g_{ij}(x(p))\) is angle preserving, that is, conformal. Two metrics which are  related by a conformal gauge transformation are called  conformally equivalent. A conformal structure does not  determine the length of any one vector at a point. Only the relative  lengths, the ratio of lengths, of any two vectors  \(\bfrac{\lvert v_p \rvert}{\lvert w_p \rvert}\) is determined. 