How to understand acting one tensor on another tensor to obtain a third tensor?

Question

I've already known the definition of the tensor that a tensor T of type $(k,l)$ is a multilinear map from a collection of dual vectors and vectors to $\mathbf{R}$: $$ T: \underbrace{T^\ast_p\times\cdots\times{}T^\ast_p}_{k\text{ times}}{\,}\times{\,}\underbrace{T_p\times\cdots\times{}T_p}_{l\text{ times}}\to \mathbf{R}$$ $"\times"$denotes the Cartesian products and $T^\ast_p$ , $T_p$ is respectively cotangent space and tangent space.

A tensor $T$ can be written as $T=T^{\mu_1\cdots\mu_k}_{\phantom{\mu_1\cdots\mu_k}\nu_1\cdots\nu_l} \hat{e}_{(\mu_1)}\otimes\cdots\otimes\hat{e}_{(\mu_k)}\otimes\hat{\theta}^{(\nu_1)}\otimes\cdots\otimes\hat{\theta}^{(\nu_l)}$.

My problem is how to understand a tensor acts on a tensor to be another tensor explicitly as the multilinear maps. For example, $$U^\mu_{\phantom{\mu}\nu}=T^{\mu\rho}_{\phantom{\mu\rho}\sigma}S^{\sigma}_{\phantom{\sigma}\rho\nu}$$ explicitly in $$U^\mu_{\phantom{\mu}\nu}\hat{e}_{(\mu)}\otimes\hat{\theta}^{(\nu)}=T^{\mu\rho}_{\phantom{\mu\rho}\sigma}\hat{e}_{(\mu)}\otimes\hat{e}_{(\rho)}\otimes\hat{\theta}^{(\sigma)}S^{\sigma}_{\phantom{\sigma}\rho\nu}\hat{e}_{(\sigma)}\otimes\hat{\theta}^{(\rho)}\otimes\hat{\theta}^{(\nu)}$$ What map does $T^{\mu\rho}_{\phantom{\mu\rho}\sigma}S^{\sigma}_{\phantom{\sigma}\rho\nu}$ precisely mean and why does it equal to the map denoted by $U^\mu_{\phantom{\mu}\nu}$ ?

Answer 1

There are two relevant operations you want to be familiar with on tensors.

First, given a $(m,n)$ tensor $T$ and an $(k,l)$ tensor $S$, you can construct their tensor product $T \otimes S$ which will be a $(m + k, n + l)$ tensor. Invariantly,

$$ (T \otimes S)(\varphi^1, \ldots, \varphi^m, \varphi^{m+1}, \ldots, \varphi^{m + k}, v_1, \ldots, v_n, v_{n+1}, \ldots, v_{n+l}) := \\ T(\varphi^1, \ldots, \varphi^m, v_1, \ldots, v_n) \cdot S(\varphi^{m+1}, \ldots, \varphi^{m+k}, v_{n+1}, \ldots, v_{n+l}). $$

If you choose a basis $(e_1, \ldots, e_n)$ for $V$ and the corresponding dual basis $(e^1, \ldots, e^n)$ for $V^{*}$ then if

$$ T = T^{\mu_1 \ldots \mu_m}_{\phantom{\mu_1 \ldots \mu_m}\nu_1 \ldots \nu_n} e_{\mu_1} \otimes \ldots \otimes e_{\mu_m} \otimes e^{\nu_1} \otimes \ldots \otimes e^{\nu_n}, \\ S = S^{\rho_1 \ldots \rho_k}_{\phantom{\rho_1 \ldots \rho_k} \sigma_1 \ldots \sigma_l} e_{\rho_1} \otimes \ldots \otimes e_{\rho_k} \otimes e^{\sigma_1} \otimes \ldots \otimes e^{\sigma_l} $$

then $ T \otimes S $ will be represented by

$$ T^{\mu_1 \ldots \mu_m}_{\phantom{\mu_1 \ldots \mu_m}\nu_1 \ldots \nu_n} S^{\rho_1 \ldots \rho_k}_{\phantom{\rho_1 \ldots \rho_k} \sigma_1 \ldots \sigma_l} e_{\mu_1} \otimes \ldots \otimes e_{\mu_m} \otimes e_{\rho_1} \otimes \ldots \otimes e_{\rho_k} \otimes e^{\nu_1} \otimes \ldots \otimes e^{\nu_n} \otimes e^{\sigma_1} \otimes \ldots \otimes e^{\sigma_l}. $$

Second, given an $(m,n)$ tensor $T$ with $m, n \geq 1$ and a choice of indices $1 \leq k \leq m$ and $1 \leq l \leq n$, you can define a $(m-1,n-1)$ tensor called the contraction or trace of $T$ with respect to $(k,l)$. To define it, choose some basis $(e_1, \ldots, e_n)$ for $V$ and define

$$ \mathrm{tr}_{(k,l)}(T)(\varphi^1, \ldots, \varphi^{m-1}, v_1, \ldots, v_{n - 1}) := T(\varphi^1, \ldots, \varphi^{k-1}, e^\sigma, \varphi^{k}, \ldots, \varphi^{m-1}, v_1, \ldots, v_{l-1}, e_\sigma, v_{l}, \ldots, v_{n - 1}). $$

You can check that this definition does not depend on the basis $(e_1, \ldots, e_n)$. In particular, if you choose a basis $(e_1, \ldots, e_n)$ for $V$ and represent $T$ as

$$ T = T^{\mu_1 \ldots \mu_m}_{\phantom{\mu_1 \ldots \mu_m}\nu_1 \ldots \nu_n} e_{\mu_1} \otimes \ldots \otimes e_{\mu_m} \otimes e^{\nu_1} \otimes \ldots \otimes e^{\nu_n} $$

then $ \mathrm{tr}_{(k,l)}(T) $ will be represented as

$$ T^{\mu_1 \ldots \mu_{k-1} \sigma \mu_{k+1} \mu_{m}}_{\phantom{\mu_1 \ldots \mu_{k-1} \sigma \mu_{k+1} \mu_{m}}\nu_1 \ldots \nu_{l-1} \sigma \nu_{l+1} \ldots \nu_n} e_{\mu_1} \otimes \ldots e_{\mu_{k-1}} \otimes e_{\mu_{k+1}} \ldots \otimes e_{\mu_m} \otimes e^{\nu_1} \otimes \ldots \otimes e^{\nu_{l-1}} \otimes e^{\nu_{l+1}} \otimes \ldots \otimes e^{\nu_n}. $$

That is, the representation of $\mathrm{tr}_{(k,l)}(T)$ is obtained by choosing a covariant and a contravariant index and "canceling" (contracting) them out using summation.

The name trace is justified by the fact that if $T$ is a $(1,1)$ tensor then you can easily check that the $(1,1)$ trace of $T$ is the same as the trace of $T$ considered as a linear map $V \to V$ (or $V^{*} \to V^{*}$).

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Returning to your question, $T$ is a $(2,1)$ tensor and $S$ is a $(1,2)$ tensor. The $(1,1)$ tensor $U$ is obtained by taking the tensor product $T \otimes S$ of $T$ and $S$ (resulting in a $(3,3)$ tensor) and then contracting twice. More explicitly,

$$ U = \mathrm{tr}_{(2,1)} (\mathrm{tr}_{(3,1)} (T \otimes S)) = \mathrm{tr}_{(2,1)} (\mathrm{tr}_{(2,2)} (T \otimes S) ).$$