nLab conformally flat manifold

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: $R\big( e^{f_i} g_{\vert U_i} \big) = 0$.

Examples

• A hyperbolic manifold is conformally flat, see there.

Related concepts

• conformal compactification

• asymptotically flat spacetime

References

See also

• Wikipedia, Conformally flat manifold

• Michael Kapovich, Conformally flat metrics on 4-manifolds, J. Differential Geom. Volume 66, Number 2 (2004), 289-301 (euclid:1102538612)