Ricci curvature tensor vanishes - (Measures deviation of volumes of “geodesic balls” from Euclidean balls of the same radius)
Ricci-flat
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Calabi-Yau
Yau, Fields Medal 1982: There are Ricci-flat but non-flat (nontrivial holonomy) projective complex manifolds of dimensions \(\geq 2\).
- CY metric