nLab conformal map

Definition

A smooth function $f \;\colon\; X_1 \to X_2$ between two Riemannian manifold $(X_1,g_1)$, $(X_2,g_2)$ is called conformal if it preserves the conformal geometry induced by the Riemannian metrics $g_1$ and $g_2$, hence if there is a smooth function $r\;\colon\; X_1 \to \mathbb{R}$ such that

$$f^\ast g_2 = r g_1 \,.$$

If $X_1 = X_2$ and $f$ is a diffeomorphism, then this is also called a conformal transformation.

Examples

• stereographic projection

• conformal compactification

References

• Wikipedia, Conformal map