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