In differential geometry, the Einstein tensor is a 2-tensor defined over Riemannian manifolds and which is defined in index-free notation as,

where is the Ricci tensor, is the metric tensor and R is the Ricci scalar (or scalar curvature). In components, the above equation reads

,

The Bianchi identities can be easily expressed with the aid of the Einstein tensor:

.

In general relativity, the Einstein tensor allows a compact expression of the Einstein equations:

.

The Bianchi identities automatically ensure the conservation of the energy-momentum tensor in curved spacetimes:

.