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Let $M$ be a manifold of dimension $n>3$ and $g$ a Riemannian metric on $M$ which is conformally equivalent to a flat one. Are there formulas (different from $R(u,v)w=\nabla_u\nabla_vw-\nabla_v\nabla_uw-\nabla_{[u,v]}w$) that allow to compute easily the curvature tensor of a conformally flat manifold such as $(M,g)$?

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A good place to start is the wikipedia article which describes how various curvatures transform under conformal change a Riemannian metric. Now, apply the formula for the transformation of the full curvature tensor to your situation, taking into account that initially you have a flat metric.

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