= Constant Ricci curvature without constant sectional curvature
{title2=$(S^2\times S^2,g_{S^2}\oplus g_{S^2})$}
The <product Riemannian metric> of two round unit spheres has <Ricci curvature> equal to its metric, since the factor Ricci tensors equal their metrics and the product connection splits. Its directional Ricci curvatures are all one. A two-plane tangent to a single sphere has <sectional curvature> one, while a mixed plane has sectional curvature zero. This four-dimensional <Einstein manifold> therefore need not have constant sectional curvature.
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