Ricci map

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Definition

The Ricci map is a map from the space of (1,3)-tensors on a manifold to the space of (0,2)-tensors, defined as follows:

Ric(F)(X,Y) = Tr(Z \mapsto F(X,Z)Y)

For instance, the Ricci map applied to the Riemann curvature tensor, gives the Ricci curvature tensor.

Facts

Riemannian curvature space

When looking at the Ricci map, we often consider its restriction on the Riemannian curvature space at each point, rather than on the space of all (1,3)-tensors. The kernel of this map is termed the Weyl curvature space and the image of this map is termed the Ricci curvature space.