Contraction of a tensor

An operation of tensor algebra that associates with a tensor with components $ a^{i_{1} \ldots i_{p}}_{j_{1} \ldots j_{q}} $, $ p,q \geq 1 $, the tensor \begin{align} b^{i_{2} \ldots i_{p}}_{j_{1} \ldots j_{q - 1}} & = a^{1 i_{2} \ldots i_{p}}_{j_{1} \ldots j_{q - 1} 1} + a^{2 i_{2} \ldots i_{p}}_{j_{1} \ldots j_{q - 1} 2} + \cdots + a^{n i_{2} \ldots i_{p}}_{j_{1} \ldots j_{q - 1} n} \\ & = a^{\alpha i_{2} \ldots i_{p}}_{j_{1} \ldots j_{q - 1} \alpha}. \end{align} (Here, the contraction is made with respect to the pair of indices $ i_{1} $ and $ j_{q} $). The contraction of a tensor with respect to any pair of upper and lower indices is defined similarly. The $ p $-fold contraction of a tensor that is $ p $-times covariant and $ p $-times contravariant is an invariant. Thus, the contraction of the tensor with components $ a^{i}_{j} $ is an invariant $ a^{i}_{i} $, called the trace of the tensor; it is denoted by $ \text{Sp}(a^{i}_{j}) $, or $ \text{tr}(a^{i}_{j}) $. A contraction of the product of two tensors is a contraction of the product with respect to an upper index of one factor and a lower index of the other.

References

[a1]	P.K. [P.K. Rashevskii] Rashewski, “Riemannsche Geometrie und Tensoranalyse”, Deutsch. Verlag Wissenschaft. (1959) (Translated from Russian)