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I am trying to understand, in as simple terms as possible:

  1. How to define integration for non-orientable manifolds, and
  2. why it is impossible to do so using only differential forms.

In particular, I've seen some discussion of using "densities" instead of $n$-forms for integration, but am not really clear on why densities are required. In other words, is it really impossible to define integration on nonorientable manifolds using forms alone?

I am of course aware that any $n$-form must vanish somewhere on a nonorientable manifold, so we cannot find a volume form, hence cannot use the standard definition of integration. I think the reason I'm not finding this answer satisfying is that it is a bit tautological: we can't define integration with respect to volume forms because there are no volume forms. But why must we define integration with respect to a (global) volume form in the first place? Is there really no other way to do it using locally-defined forms? Thinking of a manifold as a collection of local charts is common in geometry, and I'm having trouble understanding why this approach doesn't work in the case of integration.

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  • $\begingroup$ If you are defining an orientation as a class of $n$-forms for the equivalence relation given by multiplication by positive functions, then there are waaaay too many orientations. $\endgroup$ Oct 5, 2010 at 18:19
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    $\begingroup$ Why do regard this positive number you get as a "volume"? $\endgroup$ Oct 5, 2010 at 18:39
  • $\begingroup$ I didn't understand how you're going to do this procedure: "We can then use our partition of unity to "sum up" these positive values over the manifold to get a positive total volume." $\endgroup$
    – Ronaldo
    Oct 6, 2010 at 2:18
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    $\begingroup$ The reason why people define integration with respect to forms is because they want a situation where you can generalize the fundamental theorem of calculus -- which is implicitly an oriented concept, as the interval has an initial point and a terminal point. That generalization is Stokes' theorem. There are of course all kinds of other notions of integration and they're all perfectly fine. But you use forms when you want to integrate with respect to oriented volumes, not just plain old measures. $\endgroup$ Oct 6, 2010 at 17:33
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    $\begingroup$ Reading your question again, I notice you don't specify what you want to integrate. If you're interested in integrating real-valued functions then densities are precisely what you need. But if you're happy integrating other things (like differential forms) then differential forms are all you need. $\endgroup$ Apr 21, 2011 at 23:30

2 Answers 2

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Differential forms inherently measure orientation. The value of a differential form $\omega \in \bigwedge^n(M)$ on an $n$-parallelotope, i.e. $\omega(X_1, \ldots, X_n)$, is interpreted as the oriented volume of the parallelotope spanned by $X_1, \ldots, X_n$. The orientation is a necessary part of the interpretation, since differential forms are alternating: $$ \omega(X_1, X_2, \ldots, X_n) = - \omega(X_2, X_1, \ldots, X_n). $$ Hence, when working with differential forms, we should expect things to go wrong if we throw the notion of orientation out the window.

Concretely, say we want to integrate a differential form $\omega$ over the image $\phi(U)$ of a single coordinate chart. Say that $\omega$ is written in coordinates on $U$ as $\omega = a dx_1 \wedge \cdots \wedge dx_n$. The usual way to define the integral is by pulling back into Euclidean space: \begin{align*} \int_{\phi(U)}d\omega &= \int_{\phi(U)} a \, dx_1 \wedge \cdots \wedge dx_n \\ &= \int_U (\phi^* a) \, dx_1 \cdots dx_n \\ &= \int_U (a \circ \phi) \det d\phi \, dx_1 \cdots dx_n. \end{align*} This final expression involves the Jacobian determinant of the coordinate transform --- the factor picked up by change-of-variable --- and its sign depends on whether $\phi$ is orientation-preserving. Hence, if we flip orientations, we flip the sign of the integral. Hence we must have an orientation on our manifold in order for integration to be well-defined!

Clearly, if we want to be able to integrate without worrying about orientation, we either need to (a) change the definition of $\int_{\phi(U)} d \omega$, or (b) integrate against something besides differential forms. It seems you are arguing that we should try (a). But as long as you want your definition of integration to make any sense (e.g. be independent of things like choice of charts or partition of unity), by pursuing (a) you'll probably end up arriving at something that is morally more like (b), since we invented differential forms to be orientation-measuring objects in the first place. In fact, you may end up reinventing the exact concept of density that you were trying to avoid!

On that note, it may comfort you to know that any differential form yields an $s$-density $|\omega|^s$ in the natural way, as $$ |\omega|^s(X_1, \ldots, X_n) = |\omega(X_1, \ldots, X_n)|^s. $$ So, moving from differential forms to densities is really quite natural --- they're just an orientation-forgetting generalization of differential forms. The machinery involved in defining them is a bit more complicated, but that's the price we pay for dropping orientation.


Looking at it another way, it may be helpful to replace the word "possible" in your question with the word "useful". After all, any construction (that is not logically inconsistent) is possible in mathematics, but most constructions are not useful. Making that substitution:

In other words, is it really not useful to define integration on nonorientable manifolds using forms alone?

No, it's not particularly useful. See above --- orientation is baked into the definition of differential forms and their integration. Attempting to wrangle forms into playing nicely with orientation-less structures won't be pretty. We'll cause a lot more problems than we solve by trying to do that. If we want to forget about orientation, we should integrate against something else.

I've seen some discussion of using "densities" instead of n-forms for integration, but am not really clear on why densities are useful.

Densities are useful precisely because they solve the problem we're talking about here --- they are the closest thing to differential forms that we can integrate without having to worry about orientation.

I hope this clears things up!

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On an orientable manifold, we define integration of functions with respect to a volume form. On a non orientable manifold, there are no volume forms, so we have to do something else!

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    $\begingroup$ Thanks for your answer. I think the reason I'm not finding it satisfying is that it's a bit tautological: we can't define integration with respect to volume forms because there are no volume forms. But why must we define integration with respect to (global) volume forms? Is there really no other way to do it using local forms? Thinking of a manifold as a collection of local charts is common in geometry, and I'm having trouble understanding why it's impossible in the case of integration. $\endgroup$ Oct 6, 2010 at 15:46
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    $\begingroup$ @funarharpsichord: well, what do you want to do with the integral on a non-orientable manifold? The reason we define integration on manifolds as we do is because it is the only way to do the things we want to do. Maybe you could state a concrete problem that you want to solve, and then we may be more concrete. $\endgroup$ Oct 6, 2010 at 17:09
  • $\begingroup$ Forms are needed because integrals have a sign. For instance the sign of a line integral depends on the direction that the curve is parameterized. In general this idea leads to the idea of an orientation of the manifold. I you just want to compute unoriented areas, you do not strictly speaking need differential forms. I wonder if you could generaze integration of functions to unorientable manifolds by fulling the function back to the 2 fold cover, integrate there, then divide by 2. $\endgroup$
    – lavinia
    Dec 13, 2013 at 17:27

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