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What it the standard notation used while working with vector-valued differential forms?

I tried using abstract index notation, for example denoting a $1$-form valued $2$-form as $P_{i[bc]}$, but I'm not sure I'm doing it right.

What is the standard way to denote vector-valued differential forms (at least of kind $\Lambda^p(T^*M) \otimes \Lambda^n(T^*M)$), tensor and exterior products, exterior derivative, Hodge dual, and contractions of all kinds?

If you are interested why do I need all these, that's because of fluxes and balance equations, and not only for scalar quantities.

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2 Answers 2

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Abstract indices is a good alternative to work with, just don't be afraid of using different ranges of symbols dedicated to each bundle you deal with. For instance, I prefer to use the letters $a$, $b$, $c$ etc (the initial segment of the Latin alphabet) as the labels for the tangent and cotangent bundles (depending on the position), and juxtapositions of these letters denote various tensor bundles. Using parentheses and brackets we can indicate tensor parts of these bundles.

Thus, as you correctly noticed $E_{a[bc]}$ can be seen as the bundle of 2-form-valued 1-forms, and this notation is indeed standard. If there is another bundle that I want to be treated differently from the tensor bundles, I would use letters $\Phi$, $\Psi$, $\Xi$ etc associated to this bundle, and again tensor products of thus bundle and its dual will be simply denoted by using letter $E$ with the letters from the specified range attached as indices in various positions. We may have something like $E^{\Phi \Psi}{}_{\Xi}$ or $E^{[\Phi \Psi]}{}_{\Xi}$, and so on. The question is how to introduce the usual operations in this notation. This has been already done. See the references below.

For instance, the exterior covariant derivative of a 1-form with values in 2-forms will be written as simple as $2\,\nabla_{[i}P_{j][bc]}$. The notation $P_{[a}Q_{b]}$ can be used for the wedge product $P\wedge Q$ (up to the coefficient). A connection in a vector bundle $E^{\Phi}$ will be then a map $\nabla_{a}\colon E^{\Phi}\to E^{\Phi}{}_{a}$ admitting a certain sloppiness of the language. Another useful trick with the abstract indices is not always to use the indices ("suppressing") which comes naturally when one calculates in this notation a lot. For the Hodge dual you may probably want to use the volume form $\varepsilon_{a^{1}\dots a^{n}}$ which is a $n$-form with certain properties.

References:

  1. R.Penrose, W.Rindler "Spinors and space-time. Vol.1" which is usually referred to as the origin of the abstract index notation (in fact, it was known to physicists a while before the book appeared, but R.Penrose gave a logical foundation to expose its mathematical rigor in full glory).
  2. R.Wald "General Relativity", see Appendix B where one can find useful formulae for differential forms given in the abstract index notation. This book is somewhat more accessible then the previous one.
  3. R.W.R.Darling "Differential Forms and Connections" is more suitable for mathematicians, and is a useful reading as well. The author uses the classical "invariant" notation, such as $d^{\nabla}\omega$ for exterior covariant derivatives.

Edit. As requested I attempt to apply the abstract index notation to show the Leibniz rule for the case of a 1-form $v_{a}$ and a 2-form $w_{ab} = w_{[ab]}$. So, we want to show $$ (d(v \wedge w))_{abcd} = (dv \wedge w)_{abcd} + (-1)^{deg(v)}(v \wedge dw)_{abcd} $$ The definitions disentangled give $$ (v \wedge w)_{bcd} = \frac{3!}{1! \cdot 2!} v_{[b} w_{cd]} $$

$$ (d(v \wedge w))_{abcd} = 4 \nabla_{[a} (v \wedge w)_{bcd]} = \frac{4!}{1! \cdot 2!} \nabla_{[a} (v_{b} w_{cd]}) \tag{1} $$

$$ dv_{ab}=2 \nabla_{[a}v_{b]} $$

$$ (dv \wedge w)_{abcd} = \frac{4!}{2!\cdot 2!} (2 \nabla_{[a}v_{b})w_{cd]} = \frac{4!}{1!\cdot 2!} (\nabla_{[a}v_{b})w_{cd]} \tag{2} $$

$$ (dw)_{bcd}=3 \nabla_{[b} w_{cd]} $$

$$ (v \wedge dw)_{abcd} = \frac{4!}{1! \cdot 3!} (3 v_{[a} \nabla_{b} w_{cd]}) = \frac{4!}{1! \cdot 2!} ( v_{[a} \nabla_{b} w_{cd]}) \tag{3} $$

It remains to compare (1) with (2) and (3) to understand what is going on: the alternation is responsible for the sign $(-1)^{deg(v)}$ which is just $-1$ in our case.

Remark. If our forms have coefficients in vector bundles we need to take care to define what the wedge product really means for the calculations to make sense.

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I haven't yet got the last reference, so maybe the answer is there. I'm struggling to write the Leibniz rule for $d(v \land w)$ in indexes. Could you help please? You may take $v$ to be 1-forms and $w$ --- a 2-form. –  Yrogirg Sep 15 '12 at 11:45
    
@Yrogirg I added a few lines to address your query. –  Yuri Vyatkin Sep 16 '12 at 1:27
    
btw, what's the advantage of having all these coefficients in front of the wedge product? –  Yrogirg Sep 28 '12 at 13:16
    
These coefficients in the definitions make the explicit expressions have integer coefficients in their terms (not fractions). Also, these are standard conventions, see e.g. Chapter 4 of Spivak's "Calculus on manifolds". You may know that some authors use different conventions. –  Yuri Vyatkin Sep 28 '12 at 13:21
    
I meant what is the advantage of having them in the notation? That is $[\,]$ is meant to represent alternation, wouldn't it be better in all cases to denote by $[\,]$ the wedge product/form itself? –  Yrogirg Sep 28 '12 at 13:41

Let $E$ be a smooth rank $k$ vector bundle on a smooth manifold $X$. I would write an $E$-valued $p$-form (an element of $\Omega^P(E) := \Gamma(X, \bigwedge^P(T^*X)\otimes E)$) locally (i.e. on a trivialising open set for the bundle $\bigwedge^P(T^*X)\otimes E$) as $$f_I^{\ \alpha}dx^I\otimes s_{\alpha}$$ where $I$ is a $p$-multiindex and $\{s_{\alpha}\ |\ \alpha = 1,\ \dots,\ k\}$ is a basis of local smooth sections of $E$. Note, I am using Einstein summation notation.

As for extending the operations on forms, sometimes authors use the same notation, other times they will include the name of the bundle using either a superscript or a subscript. For example, the Hodge star acting on $E$-valued forms is sometimes written as $\ast_E$.

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Why have you introduced bases? Is it really necessary? What if I also need $E$ dual $E^*$? –  Yrogirg Sep 10 '12 at 12:39
1  
It all transforms as you expect, so you can pretend it's abstract index notation if you like. –  Zhen Lin Sep 10 '12 at 12:48
    
Zhen is right. I am using Einstein summation notation and I think you are using abstract index notation, in which case you would forget about the $dx^I\otimes s_{\alpha}$ and just write $f_I^{\alpha}$. –  Michael Albanese Sep 10 '12 at 12:54
    
that's good, but what's with duals? How will you distinguish $E$ and $E^*$? Usually it's done with upper and lower indices. –  Yrogirg Sep 10 '12 at 13:03
4  
Different alphabets. Note that indices related to the coordinate basis are written in Latin while indices related to the bundle frame are written in Greek. –  Zhen Lin Sep 10 '12 at 13:15

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