# Riemannian curvature space

From Diffgeom

## Definition

Let be a real vector space with a Euclidean inner product. The **Riemannian curvature space** of is the space of -tensors on of the following description:

is in the Riemannian curvature space if is alternating in and , and further, for any , the map is alternating in and .

By the canonical identification of -tensors with -tensors, the Riemannian curvature space can be identified with the symmetric square of the exterior square of the vector space.