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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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    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.2010-10-05
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    Why do regard this positive number you get as a "volume"?2010-10-05
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    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."2010-10-06
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    Another thing: In this approach, it seems that the volume of the manifold depends on what family of functions f_\alpha you choose.2010-10-06
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    @Mariano: I believe my definition is correct -- here's the original one from Abraham, Marsden & Ratiu: "Two volume forms $\mu_1$ and $\mu_2$ are called equivalent if there is an $f \in \mathcal{F}(M)$ with $f(m)>0$ for all $m \in M$ such that $\mu_1 = f\mu_2$. An orientation of $M$ is an equivalence class $[\mu]$ of volume forms on $M$."2010-10-06
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    @Ronaldo: Why? To make a very crude argument: if I pick different $f_\alpha$ then I may end up integrating over a larger region in $R^n$ (say), but I'm also "spreading out" the integrand over a larger area. And this spreading out is encoded by the pushforward. A more rigorous version of this argument is made in Abraham, Marsden & Ratiu, Theorem 8.1.2.2010-10-06
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    @Robin: Because I'm trying to define a notion of volume for a nonorientable manifold. On an (abstract) orientable manifold, I can pick an arbitrary volume form, which effectively prescribes the total volume. On a nonorientable manifold I'm attempting to do a similar thing by picking a volume form in each chart. Admittedly, the "volume" I get will also depend on my choice of partition functions $g_\alpha$.2010-10-06
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    @funarharpsichord: but that definition you quote applies to *volume forms*, which do not vanish anywhere. If you have a non-orientable manifold, then there are no volume forms, and then there is no orientation on it (because there are no equivalence classes of volume forms, because there are no volume forms!)2010-10-06
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    @funarharpsichord: if you are happy with your "volume" of a non-orientable manifold to depend on your choice of partition functions, then you can much more easily define the volume of a non-orientable manifold to be $14$: you can always pick a covering and a subordinate partition of unity so that your formula adds up to $14$...2010-10-06
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    @Ronaldo: what I mean by "sum up" is to apply the same procedure used to define integration on an orientable manifold, i.e., $\int_M \omega = \sum_\alpha \int_{f_\alpha(U_\alpha)} (f_\alpha)_* (g_\alpha \omega_\alpha).$2010-10-06
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    @Mariano: correct -- but I do not apply this definition of orientation to $n$-forms on $M$, just to $n$-forms on each of the $U_\alpha$ (viewed as submanifolds of $M$). (The $U_\alpha$ are of course orientable since they each have the topology of a disk.)2010-10-06
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    Is a density on a manifold the same thing as a positive measure which is absolutely continuous with respect to Lebesgue measure on all coordinate patches and with a smooth Radon-Nikodym derivative?2010-10-06
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    @Mariano: ok, I see the confusion -- I say that I'm picking $\omega_\alpha$ on $M$. Well, imagine that $\omega_\alpha$ is not a volume form on $M$ but is a volume form when restricted to $U_\alpha$ (i.e., nonvanishing on $U_\alpha$).2010-10-06
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    @Mariano: all I know about densities is the definition given in Lee, i.e., it really is a function on a product of vector (or tangent) spaces. I'm assuming everything is smooth.2010-10-06
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    To be honest, the more you explain the less I understand what you are trying to achieve. Maybe you could edit the question and make it more explicit? In particular, move the information from the comments to the actual text.2010-10-06
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    @all: Again, I'm sorry for being imprecise / giving strange or contradictory definitions. My intention in posting here was to learn how this should really work, not make up strange new math! Still, I thought I should try a little harder than to simply ask, "how do you define integration on non-orientable manifolds, and why must it differ from the usual definition?" But that's the question I'm really interested in.2010-10-06
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    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.2010-10-06
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    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.2011-04-21

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