# nLab conformally flat manifold

Contents

### Context

#### Riemannian geometry

Riemannian geometry

# Contents

## Idea

A Riemannian manifold $(X,g)$ is conformally flat if it is locally taken to a flat manifold by a conformal transformation; more specifically, if there exists an open cover $\{U_i \to X\}_{i \in I}$ and on each $U_i$ a smooth function $f_i$, such that $e^{f_i} g_{\vert U_i}$ has vanishing Riemann curvature: $R\big( e^{f_i} g_{\vert U_i} \big) = 0$.