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Setup:

Let $M^n$ be an oriented manifold and let $$\mathcal{A} := \{(U_i,\varphi_i):i \in I\}$$ be a positively oriented atlas ($\varphi_i:U_i \to \varphi_i(U_i)$ preserves orientation). Furthermore, let $\{\chi_i\}_{i \in I}$ be a partition of unity subordinate to $\mathcal{A}$.

Then for $\omega \in \Omega^n(M)$ with compact support, we define $$\int_{M} \omega = \sum_{i \in I} \int_{\varphi_i(U_i)} (\varphi^{-1})^{*}(\chi_i \omega).$$

Now, my question is this. Exactly what do we mean by $\varphi_i:U_i \to \varphi_i(U_i)$ being "orientation-preserving" here?

My general understanding is that if we have two smooth manifolds $M^m,N^n$ with orientation $\omega_M \in \Omega^n(M): \omega_M(p) \neq 0, \forall p \in M$ and similarly for $\omega_N \in \Omega^n(N)$, we get two smooth oriented manifolds $(M,\omega_M),(N,\omega_N)$. Let´s say we have a diffeomorpism $\varphi \in C^{\infty}(M,N)$. Then we say that $\varphi$ is orientation-preserving if there exists a positive function $f \in C^{\infty}(M)$ such that $(\varphi)^{*} \omega_N = f \cdot \omega_M$

Now, in the setup above, $\varphi_i:U_i \subset M \to \varphi_i(U_i) \subset \mathbb{R}^n$, atleast if we assume $M$ is without boundary. Then, since $\varphi_i$ is a homeomorphism, $\varphi_i(U_i)$ is open in $\mathbb{R}^n$. And we know that for open sets $U \subset \mathbb{R}^n$ we get a canonical orientation (non-zero $n$-form) in coordinates on the form $$dx^1 \wedge \cdots \wedge dx^n.$$

Does this mean that what we are actually saying is that for $\varphi_i:U_i \to \varphi_i(U_i)$ to be orientation preserving, we need that for each $i \in I$, there exists a positive function $f \in C^{\infty}(M)$ such that $$(\varphi_i)^{*} (dx^1 \wedge \cdots \wedge dx^n) = f \cdot \omega_M?$$

Comment: Here, I just view $\mathbb{R}^n$ as a smooth manifold with the smooth identity structure, i.e. $(\mathbb{R}^n,[\operatorname{id}])$.

Also, $(\varphi_i)^{*}$ just denotes the pullback here.

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  • $\begingroup$ Yes, that’s it (well since we’re speaking of $U_i$, we should technically say there is a function $f\in C^{\infty}(U_i)$ such that $\phi_i^*(dx^1\wedge\dots\wedge dx^n)=f\cdot \iota_{U_i}^*(\omega_M)$, where the latter is the pullback of $\omega_M$ via the inclusion $U_i\hookrightarrow M$). $\endgroup$
    – peek-a-boo
    Aug 9, 2023 at 18:25
  • $\begingroup$ Thank you, make this into an answer and I´ll accept it, if you want to. :) $\endgroup$
    – Ben123
    Aug 9, 2023 at 18:29
  • $\begingroup$ you can go ahead and answer your own question (since it’s a simple definition really:) $\endgroup$
    – peek-a-boo
    Aug 9, 2023 at 18:30
  • $\begingroup$ Hm, I have one more reservation though. In the statement of what it means for a diffeomorphism $\varphi$ between manifolds to be orientation-preserving, the positive function $f$ should not be different depending on which point $p \in M$ we are looking at. When you say $f \in C^{\infty}(U_i)$, do you actually mean $f|_{U_i}$, so basically, it is the same function $f$, but we are just making a restrictions to different domains $U_i$? $\endgroup$
    – Ben123
    Aug 9, 2023 at 18:39
  • $\begingroup$ No, I mean for each $i$, there is a positive function $f$ defined on $U_i$. The order of quantifiers is already clear, but to be explicit, we can say “there is a positive function $f_i\in C^{\infty}(U_i)$”, indicating that it’s just a smooth positive (lol looking back I forgot this important word) function, one for each $i$, and I’m not requiring that it be the restriction to $U_i$ of some smooth positive function $f:M\to\Bbb{R}$. Notice that we’re speaking of a diffeo $\phi_i:U_i\to\phi(U_i)$, so of course we expect a different $f_i$. $\endgroup$
    – peek-a-boo
    Aug 9, 2023 at 18:42

1 Answer 1

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By input from peek-a-boo, the overall idea in my preliminary musings are correct, with some details that need to be corrected. Following peek-a-boo:s suggestion, we should say that what we mean by $\varphi_i:U_i \to \varphi_i(U_i)$ being orientation-preserving in the setup-above, is that there exists a smooth function $f \in C^{\infty}(U_i)$ such that $$(\varphi_i)^{*}(dx^1 \wedge \cdots \wedge dx^n) = f \cdot (\iota_{U_i})^{*} (\omega_M)$$ where $\iota_{U_i}:U_i \hookrightarrow M$ is the inclusion from $U_i$ to $M$.

I might edit this answer more later.

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