 Three-dimensional curvature from the Ricci tensor

ID: three-dimensional-curvature-from-the-ricci-tensor

R_{ij pq} = g_{i p} S_{j q} - g_{i q} S_{j p} - g_{j p} S_{i q} + g_{j q} S_{i p}

, where

S

is the Schouten tensor. Consequently every three-dimensional Ricci-flat Riemannian manifold is a flat Riemannian manifold. The implication fails in higher dimensions.

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