Suppose, there is a tensor of rank three, given by $C^{ij}_k$ and it is a product of two other tensors. The relation is as follows:
$C^{ij}_k$ = $A^{ij}$$B_k$
This $C^{ij}_k$ has 27 elements in it.
If I contract the indices $j$ and $k$ Then I get the following:
$C^i$ = $A^{ij}$$B_j$
By equating $j$ and $k$ the tensor has been reduced to a rank 1 tensor and has only 3 elements. The process is now similar to ordinary matrix multiplication between a 3X3 and a 3X1 matrices.
I get that by equating two indices, we are selecting only a few of the elements of original $C^{ij}_k$ and summing them up, which eventually represents a vector.
My question is, what happens to the other elements? Do they become zero for some reason? If so, what property makes them zero? If they are not zero, why we are allowed to simply ignore them? what do they represent?