# 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 and be two Riemannian metrics on a differential manifold . Then we say that is conformally equivalent to if there is a scalar function such that for any , and tangent vectors :