# nLab conformal map

### Context

#### Riemannian geometry

Riemannian geometry

## Basic definitions

• Riemannian manifold

• moduli space of Riemannian metrics

• pseudo-Riemannian manifold

• geodesic

• Levi-Civita connection

• ## Theorems

• Poincaré conjecture-theorem
• ## Applications

• gravity

• # Contents

## 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.

## References

Last revised on May 1, 2017 at 06:46:02. See the history of this page for a list of all contributions to it.