Notation and naming for two operations with $p$-form valued $n$-forms

Question

While trying to answer my other question I found I never heard about vector-valued differential forms. I've been searching for them in various mathematical physics books, but didn't get too much.

I'm interested in notion and names of two operations with $p$-form valued $n$-forms.

I treat p-form valued n-form as

$$ (TM)^n \to \Omega^p(M) $$

or

$$ (TM)^n \to (TM)^p \to \mathbb R $$

I think ususal contraction with a vector field is denoted as $\rfloor$:

$$\rfloor : TM \to \left(TM^n \to TM^p \to \mathbb R \right) \to \left(TM^{n-1} \to TM^{p} \to \mathbb R \right) $$

How do I denote the similar operation, but when contracting with the resulting $p$-form? :

$$\lfloor : TM \to \left(TM^n \to TM^p \to \mathbb R \right) \to \left(TM^n \to TM^{p-1} \to \mathbb R \right) $$

Or maybe there is a notation for switching position, that is $p$-form valued $n$-forms is turned into $n$-form valued $p$-forms.

Given an $n$-form and a $p$-form I can produce $p$-form valued $n$-form. How do I denote and name this operation?

$$(TM^n \to \mathbb R) \to (TM^p \to \mathbb R) \to \left( TM^n \to TM^p \to \mathbb R \right)$$

$$(\omega,\nu) \mapsto \left(\left(v^n,u^p\right) \mapsto \omega(v^n) \nu(u^p)\right)$$

Maybe these operations are not specific only to this setting and have some universal nature and notation.

Answer 1

If $E \to M$ is a vector bundle then $E$-valued $n$-forms are sections of the tensor product bundle $E \otimes \Lambda^n(T^*M)$. From basic properties of the tensor product of bundles, sections are spanned by $\sigma \otimes \mu$ where $\sigma \in \Gamma(E), \mu \in \Omega^n(M)$. In your situation, you're looking at sections of the bundle $\Lambda^p(T^*M) \otimes \Lambda^n(T^*M)$ so the last construction you mention is just tensoring a section of $\Lambda^p(T^*M)$ with one of $\Lambda^n(T^*M)$ to get a section of the tensor product of the bundles.

As for contraction, this is from the general fact that any bundle map $E \to F$ extends to a bundle map $E \otimes \Lambda^n(T^*M) \to F \otimes \Lambda^n(T^*M)$ by acting as the identity on the second factor of the tensor product. In your situation $E = \Lambda^p(T^* M)$ and $F = \Lambda^{p-1}(T^* M)$.