# 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.