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Gaussian curvature, mean curvature, sectional curvature, scalar curvature, curvature tensors (Riemann, Ricci, Weyl)

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Can anyone give an example of Ricci flat Riemannian or Lorentzian Manifold that is not flat?

Eguchi - Hanson metric over $T^*S^2$ is complete Ricci flat but not flat, which can be written down explicitly. In fact, $T^*S^n$ admits Calabi-Yau structure for each $n$. Ref: Stenzel, Ricci flat m …
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