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 Nov 13 awarded Commentator Nov 13 comment How do you show that the Laplacian is the square of the (Euclidean) Dirac operator? Yes Marek, that is exactly what I'm forgetting. Thanks for pointing out this obvious blunder! Nov 13 asked How do you show that the Laplacian is the square of the (Euclidean) Dirac operator? Oct 6 revised Is the notion of density really needed to define integration on nonorientable manifolds? deleted 1 characters in body Oct 6 revised Is the notion of density really needed to define integration on nonorientable manifolds? added 391 characters in body; added 3 characters in body; added 23 characters in body Oct 6 comment Is the notion of density really needed to define integration on nonorientable manifolds? 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. Oct 6 awarded Editor Oct 6 revised Is the notion of density really needed to define integration on nonorientable manifolds? deleted 2348 characters in body; added 290 characters in body Oct 6 comment Is the notion of density really needed to define integration on nonorientable manifolds? @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. Oct 6 comment Is the notion of density really needed to define integration on nonorientable manifolds? @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. Oct 6 comment Is the notion of density really needed to define integration on nonorientable manifolds? @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 $U_\alpha$). Oct 6 comment Is the notion of density really needed to define integration on nonorientable manifolds? @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.) Oct 6 comment Is the notion of density really needed to define integration on nonorientable manifolds? @Ronaldo: what I mean by "sum up" is to apply the same procedure used to define integration on an orientable manifold, i.e., $\int_M \omega = \sum_\alpha \int_{f_\alpha(U_\alpha)} (f_\alpha)_* (g_\alpha \omega_\alpha).$ Oct 6 comment Is the notion of density really needed to define integration on nonorientable manifolds? @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 $g_\alpha$. Oct 6 comment Is the notion of density really needed to define integration on nonorientable manifolds? @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. Oct 6 comment Is the notion of density really needed to define integration on nonorientable manifolds? @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 $M$." Oct 5 awarded Student Oct 5 asked Is the notion of density really needed to define integration on nonorientable manifolds?