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Examples of signature (2,2) manifolds with commuting curvature operators
Miguel Brozos-Vazquez, Eduardo Garcia-Rio, Peter B. Gilkey and Ramon Vazquez-Lorenzo
We exhibit Walker manifolds of signature (2,2) with various commutativity properties for the Ricci operator, the skew-symmetric curvature operator, and the Jacobi operator. If the Walker metric is a Riemannian extension of an underlying affine structure, these properties are related to the Ricci tensor of the affine structure.