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 $f:M\to \mathbb {R} ^{*}$ such that for any $p\in M$, and tangent vectors $v,w\in T_{p}M$:

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