# Ricci map

From Diffgeom

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## Definition

The **Ricci map** is a map from the space of -tensors on a manifold to the space of -tensors, defined as follows:

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 -tensors. The kernel of this map is termed the Weyl curvature space and the image of this map is termed the Ricci curvature space.