Conformal map

A priori, this expression has two uses:

In complex analysis

It is a holomorphic function such that $f^{\prime }(z)\ne 0$.

In Riemannian geometry

Let $(M,g)$ be a pseudo-Riemannian manifold. For any smooth function $\rho$ the metric $\tilde{g}=e^{2\rho }g$ is said to be conformal or conformally related to $g$.

Let $F$ a smooth map from $(M,g)$ to another Riemannian manifold $(N,h)$. If the Riemannian metric $F^{\ast }(h)$ induced on $M$ is conformal to the original $g$, then $F$ is called a conformal mapping of $(M,g)$ to $(N,h)$.

Under a conformal mapping the angle between two vectors is preserved.

A Riemannian manifold is said to be conformally flat if for every point there exists a neighbourhood and a conformal map from it to ${\mathbb{R}}^n$ with the standard metric. In the case of surfaces, it is always the case due to the existence of isothermal coordinates.