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When studying fibre bundles, connections and gauge theories it is usual to consider vector-valued differential forms, like the connection one-form, or it's pull back by a local trivialization known as the gauge potential. In those examples, the form takes value on the Lie algebra of the group, and hence, they take values on some vector space $V$.

Now, for regular $\mathbb{R}$-valued differential forms, integration is simple to define. If $c$ is a singular $k$-cube and $\omega$ a differential $k$-form, then

$$\int_{c} \omega = \int_{[0,1]^k} c^\ast \omega.$$

Now, how the integral of a $V$-valued differential form is defined? Is it simply componentwise? If so, I'm unsure if it depends on the basis chosen for the vector space.

So how one deals with this kind of integration? Is it some well-defined concept?

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It's just componentwise. Fix a basis $e_i$ with dual basis $\theta^i$ and define $$\int \omega = \sum_i \left(\int \theta^i \circ \omega\right) e_i.$$

If you have another basis $v_i$ with dual $\pi^i$, then let $A$ be the change of basis matrix such that $v_j = \sum_i A_{ji} e_i$. A little bit of linear algebra tells us that the dual bases are then related by $\pi^j = \sum_i A^{-1}_{ij} \theta^i$, so by the linearity of integration we have

$$ \sum_i \left( \int \pi^i \circ \omega \right) v_i = \sum_{i,j} A^{-1}_{ji} \left( \int \theta^j \circ \omega \right) v_i =\sum_{i,j,k} A^{-1}_{ji} A_{ik} \left( \int \theta^j \circ \omega \right) e_k = \sum_{j} \left( \int \theta^j \circ \omega \right) e_j;$$ i.e. the integral is basis-independent.

If you want a more abstract definition then something like the Bochner integral might be more satisfying, but it's not at all necessary in finite dimension.

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  • $\begingroup$ This appears to only be accurate for coordinate transformations that are coordinate independent. Am I missing something here? $\endgroup$
    – juacala
    May 24 '20 at 20:29
  • $\begingroup$ @juacala: are you saying you expect the $A_{ij}$ to depend on the position in the base manifold? The forms we're considering here take values in a fixed vector space, not a vector bundle, so our bases $e,v$ are fixed, not varying frames. $\endgroup$ May 24 '20 at 22:49
  • $\begingroup$ I guess I missed something from the question. Certainly in the context of General Relativity in curved space-time, you can't meaningfully integrate a vector-valued form over a region in the manifold. This is why "pseudo-tensors" were invented to create some stress-energy-like object that is integrable over the manifold. $\endgroup$
    – juacala
    Jun 11 '20 at 15:53
  • $\begingroup$ @juacala: The gravitational stress-energy problem in GR is a physics problem, not a maths one: you can't define a gravitational stress-energy that is both coordinate-invariant and physically meaningful. If you're already given a 4-form on a 4-manifold, you can absolutely integrate it (this capability essentially defines differential forms!), and if it takes values in some vector space then the integral will also. $\endgroup$ Jun 11 '20 at 23:42
  • $\begingroup$ Perhaps this is an issue with terminology-mismatch, but I definitely think it's a maths problem: see mathoverflow.net/questions/35334/… (the problem described in this question is exactly the problem of integrating the stress-energy tensor) $\endgroup$
    – juacala
    Jun 12 '20 at 21:34
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When use integral to calculate something like mass $M$ or charge $Q$ of an object, we assume that we can define mass or charge density $\rho:\mathbb R^3,\to \mathbb R$ as a real valued function of space (coordinate), and the sum of step function mean the summable of mass of space regions which intersect zero volume measure.

When use integral to calculate something like electrostatic field via charge density, we use coulomb's law and superposition principle. In this times, a point $x_0\in \mathbb R^3$ in space produce a unit volume electric field density $x\mapsto \frac{k\rho(x_0)}{|x-x_0|^2}\frac{x-x_0}{|x-x_0|}$, i.e. a point $x_0$ in space maps to a vector field define on the whole space (consider singularity). We can calculate electric field at every point in space by superposition principle.

So I have some rough thought: for vector bundle on manifold, we can mimic electrostatic field, talk about the integration of vector field valued differential form. (or call it section of vector bundle value).

And note that (1) it does not use connection or paralled transpot on the vector bundle. (2) paralled transport depend on path, unless the connection is flat, or curvature is zero. (3) If we have matrix value form (take value in $V^*_p\otimes V_p$ at point $p$), such as curvature matrix, then use trace or determinant of matrix will give you a real or compex valued form.

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