Riemannian curvature space

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Definition

Let V be a real vector space with a Euclidean inner product. The Riemannian curvature space of V is the space of (1,3)-tensors on V of the following description:

R is in the Riemannian curvature space if R(X,Y,Z) is alternating in X and Y, and further, for any X,Y, the map (Z,W) \mapsto g(R(X,Y,Z),W) is alternating in Z and W.

By the canonical identification of (1,3)-tensors with (0,4)-tensors, the Riemannian curvature space can be identified with the symmetric square of the exterior square of the vector space.