# Definition:Pointwise Conformal Riemannian Metrics

## Definition

Let $M$ be a Riemannian manifold with metrics $g_1$ and $g_2$.

Let $f \in \map {C^\infty} M$ be a positive smooth real function.

Suppose:

$g_1 = f g_2$

where $f g_2$ denotes the scalar multiplication of $g_2$ by $f$.

Then $g_1$ and $g_2$ are said to be pointwise conformal to each other.