Geometric realizations of generalized algebraic curvature operators
  We study the 8 natural GL equivariant geometric realization questions for the space of generalized algebraic curvature tensors. All but one of them is solvable; a non-zero projectively flat Ricci antisymmetric generalized algebraic curvature is not geometrically realizable by a projectively flat Ricci antisymmetric torsion free connection. 
P. Gilkey, S. Nikcevic, and D. Westerman