Conformally equivalent metrics

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Template:Riemannian metric relation

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_pM:

g_1(v,w) = f(p)g_2(v,w)