If $A^iB_i$ is called a contraction, what is $A^{ij}B_{ij}$ called?

Question

I have a tensor $A^{ijk}$ and a tensor $B_{ijk}$, and I'd like to contract all the indices between them, resulting in the scalar $k$:

$$k = A^{ijk}B_{ijk}$$

Is there a name for this sort of "multi-contraction"?

By knowing the name, I'd like to study this type of operation more. I'm not sure whether they occur in the study of differential forms, or tensor algebra, or geometric algebra, or otherwise.

Answer 1

They occur in many places. An example is the scalar curvature

$$ S = g^{ij}R_{ij} $$

where $g^{ij}$ is the metric tensor and $R^{ij}$ the Ricci curvature, which in turn is also the contraction of the Riemann curvature tensor