View source for Contraction of a tensor

An operation of tensor algebra that associates with a tensor with components <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025870/c0258701.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025870/c0258702.png" />, the tensor

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(here the contraction is made with respect to the pair of indices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025870/c0258706.png" />). The contraction of a tensor with respect to any pair of upper and lower indices is defined similarly. The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025870/c0258707.png" />-fold contraction of a tensor that is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025870/c0258708.png" />-times covariant and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025870/c0258709.png" />-times contravariant is an invariant. Thus, the contraction of the tensor with components <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025870/c02587010.png" /> is an invariant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025870/c02587011.png" />, called the trace of the tensor; it is denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025870/c02587012.png" />, or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/c/c025/c025870/c02587013.png" />. 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.



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====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  P.K. [P.K. Rashevskii] Rashewski,   "Riemannsche Geometrie und Tensoranalyse" , Deutsch. Verlag Wissenschaft.  (1959)  (Translated from Russian)</TD></TR></table>