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We examine questions of geometric realizability for algebraic structureswhich arise naturally in affine and Riemannian geometry. Suppose given analgebraic curvature operator R at a point P of a manifold M and suppose given areal analytic (resp. C-k for finite k at least 2) pseudo-Riemannian metric on Mdefined near P. We construct a torsion free real analytic (resp. C-k)connection D which is defined near P on the tangent bundle of M whose curvatureoperator is the given operator R at P and so that D has constant scalarcurvature. We show that if R is Ricci symmetric, then D can be chosen to beRicci symmetric; if R has trace free Ricci tensor, then D can be chosen to havetrace free Ricci tensor; if R is Ricci alternating, then D can be chosen to beRicci alternating.

Riemannian geometric realizations for Ricci tensors of generalized algebraic curvature operators