Jtdylktuy

I am currently working on implementing the inner(scalar or dot) product of two tensors of arbitrary order. As far as i understand, you need to make sure, that the last dimension of the first tensor $pmb A$ matches in size with the first dimension of the second tensor $pmb B$. The resulting tensor $pmb C$ then has order r + q - 2, if the first has order r and the second one has order q.

Is the product then just

$V_{ij...km} = T_{ij....l}, U_{l.....km} ,$ ?

I think you need to review some of the basic properties of tensors.

Firstly, all of the indices on a tensor have the same "dimension", which is the dimension of the underlying vector space on which the tensor acts - if the first index can take values in ${1,2,dots,n}$ then so can each of the other indices. An order 2 tensor can be represented (relative to a given basis) by a matrix but this will always be a square matrix.

Secondly, summing or "contracting" over a pair of indices only produces another tensor if one index is a contravariant index and the other is a covariant index. Contravariant indices are traditionally written as "upper" or superscript indices; covariant indices are traditionally written as "lower" or subscript indices. Thus

$T^{ijdots l}U_{l dots km}$

which is shorthand for

$displaystyle sum_{l=1}^n T^{ijdots l}U_{l dots km}$

is a tensor but

$T_{ijdots l}U_{l dots km}$

is not a tensor. You can certainly calculate the values or "components" of $T_{ijdots l}U_{l dots km}$ relative to a given basis, but these values will not transform in a consistent way when you change basis, so $T_{ijdots l}U_{l dots km}$ does not represent a physically or geometrically meaningful object. This would be similar to adding up the components of a vector relative to a given basis - you can do the arithmetic, but the resulting quantity is not geometrically meaningful.

You can, however, "raise" and "lower" indices using the metric tensor $g_{ij}$. So if you wanted to contract the final index of $T_{ijdots l}$ with the first index of $U_{l dots km}$ to create a tensor, you could use the metric tensor as follows:

$g^{lp}T_{ijdots p}U_{l dots km} = T_{ijdots}^{quad l} U_{l dots km} $

which is a tensor because you are contracting a contravariant index with a covariant index.

If $T_i$ and $U_p$ are two rank-one-tensors then $$g^{ij}T_iU_j,$$ is their inner product.

For rank-two tensors if $T$ has components $T_{ij}$ and $U$ has components $U_{pq}$ then the rank-four tensor $Totimes U$ has components $(Totimes U)_{ijpq}:=T_{ij}U_{pq}$.
But for the inner product $Tcdot U$ between them, which should be a scalar, this is defined as $$g^{is}g^{jt}T_{ij}U_{st},$$
note the pattern of contractions!
One can easily generalize to arbitrary rank if both of $T$ and $U$ have equal rank.

For the case of unequal ranks it is took $Tcdot U:=0$.