Tensor contraction

A tensor contraction over two paired indices. For real second-order tensors, it gives the displayed Frobenius inner product. For a fourth-order elasticity tensor and a second-order infinitesimal strain tensor, $(C:ϵ{)}_{ij}={\sum }_{kl}{C}_{ijkl}{ϵ}_{kl}$ remains a second-order tensor; double contraction need not always give a scalar.

The metric trace contracts two indices of a covariant rank-two tensor: $T=g^{\mu \nu }{T}_{\mu \nu }$. For the metric itself in $D$ dimensions, $g^{\mu \nu }{g}_{\mu \nu }=D$.