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I wonder why in books about differential geometry there seem to be no relevant results about properties of mixed differential forms like

$$\omega_{12}\ \mathbb{d}x^1\land\mathbb{d}x^2+\lambda_{123}\ \mathbb{d}x^1\land\mathbb{d}x^2\land\mathbb{d}x^3,$$

where the "+" is understood from the "$\oplus$"-construction of vector spaces. Given the second point I make below, the perspective seems to be a little inconsistent.

Firstly, sure, you can't just integrate over these kind of forms in a direct way, but it's still a reasonable vector of the exteriour algebra. Is it that you can, maybe by means of linear algebra, just a priori say that all results you might have about these objects decompose into the direct sum of the seperate results?

But moreover, general cohomology theory theory is concerned with different operators $\mathbb{d}_i$ from space to space. In deRham cohomology, which is I think the primary motivation, one doesn't really seperate the different $\mathbb{d}_i$'s going from 1-forms to 2-forms and 2-formsto 3-forms, notationally. And well, in fact even if you formally introduce different $\mathbb{d}_i$'s, you could then just use them to define the action of one big $\mathbb{d}_i$ in the direct sum space, via the action of the small $\mathbb{d}_i$'s. Is it more than a notational decission that, in general cohomolgy theory, you don't just look at a direct sum space?

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actually, I don't see someone really consider the sum of forms of different order. –  lee Jan 12 '13 at 15:23

1 Answer 1

In geometric algebra, we work with such objects all the time. A good example of such an object is a spinor. A spinor on a 3d space looks like

$$\psi = \psi^0 + \psi^{12} e_1 \wedge e_2 + \psi^{23} e_2 \wedge e_3 + \psi^{31} e_3 \wedge e_1$$

This is essentially a quaternion. The corresponding construction on a 2d space gives a 2-component object, which is essentially a complex number.

Yes, linear algebra suggests that any linear operator distributes over addition of differently-graded objects.

In geometric calculus, the exterior derivative of a multivector field $A$ is called $\nabla \wedge A$, and it also distributes over addition of differently-graded components of $A$. The generalized Stokes theorem, however, is quite a bit stronger than what differential forms would usually tell you. In geometric calculus, we write it as

$$\int_{\partial V} \underline L(dS) = \int_V \dot{\underline L}(\dot \nabla \cdot dV)$$

Where the overdots denote that the linear operator $\underline L$ is what is being differentiated. Let $\underline L(a) = \psi a$ for some spinor $\psi$. The result is

$$\int_{\partial V}\psi \, dS = \int_V (\dot \psi \dot \nabla) \, dV$$

$\nabla \psi$ includes both exterior derivatives and coderivatives. When $\psi$ is a mixed-graded object, this means the two no longer cleanly separate, which is one reason why (I suspect) mixed-graded objects are not typically considered. Explicitly, let's consider just grade-2 part of this integral. Let $\psi^0 = s$ and $\psi - \psi^0 = B$, the bivector part.

$$\int_{\partial V} s \, dS + \int_{\partial V} B \times dS = \int_V (\nabla \wedge s - \nabla \cdot B) \, dV$$

As you can see, the interior and exterior derivatives no longer decouple.

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