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 $f$.

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

Sources

• 2018: John M. Lee: Introduction to Riemannian Manifolds (2nd ed.) ... (previous) ... (next): $\S 3$: Model Riemannian Manifolds. Euclidean Spaces