Conformally equivalent metrics

Definition

Symbol-free definition

Two Riemannian metrics on a differential manifold are termed conformally equivalent if one of them can be obtained as a scalar function times the other one. In other words, at each point, one metric is simply a constant times the other metric (the constant may vary from point to point).

Definition with symbols

Let $g_{1}$ and $g_{2}$ be two Riemannian metrics on a differential manifold $M$ . Then we say that $g_{1}$ is conformally equivalent to $g_{2}$ if there is a scalar function $f : M \to R^{*}$ such that for any $p \in M$ , and tangent vectors $v, w \in T_{p} M$ :

$g_{1} (v, w) = f (p) g_{2} (v, w)$