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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.

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en.wikipedia.org/wiki/Vector-valued_differential_form In your case, the values are in a vector bundle rather than a fixed vector space elsewhere. –  Will Jagy Aug 14 '12 at 18:47
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I found a bunch of things by googling vector bundle. I learned such things from Chern, but I can't say that he ever wrote up a primer on vector bundle operations. Your best bet is probably the book Semi-Riemannian Geometry by Barrett O'Neill books.google.com/… –  Will Jagy Aug 14 '12 at 19:10

1 Answer 1

up vote 1 down vote accepted

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 $F = \Lambda^{p-1}(T^* M)$.

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Ok, I got the last, indeed it is just $\otimes$. But what the notation for contraction? The thing is natural, it easy to understand, but the question is about notation. I'm not terribly aware in notations in vector algebra, what symbols should I use? –  Yrogirg Aug 21 '12 at 13:19

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