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I think that the Riemann tensor is zero only in the presence of a flat metric. However, the Ricci Tensor and the Ricci Scalar, are unknown to me, whether they are zero only in the presence of a flat metric. Of the three tensors, Riemann Tensor, Ricci Tensor, and Ricci Scalar, which ones are only zero in a flat metric?

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If the metric is flat, of course, they're all zero, because $\text{Riemann}=0$ $\implies$ $\text{Ricci}=0$ $\implies$ $\text{scalar}=0$. But the converse depends on the dimension. In dimension $1$, every metric is flat, and the Riemann, Ricci, and scalar curvatures are always zero. In dimension $2$, if the scalar curvature is zero, the metric is flat. In dimension $3$, if the Ricci curvature is zero, the metric is flat, but there are non-flat metrics with zero scalar curvature. In dimensions $4$ and up, there are plenty of examples of non-flat metrics with both Ricci and scalar curvatures equal to zero (K3 surfaces in algebraic geometry, for example).

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