# Existence of oriented charts in the definition of the integral of a differential form

I'm reading Lee's Introduction to Smooth Manifolds, and his definition of the integral of a differential form is a bit hard for me to understand.

Suppose $$M$$ is an oriented smooth $$n$$-manifold with or without boundary, and $$\omega$$ is a compactly supported $$n$$-form on $$M$$. Let $$\{U_i\}$$ be a finite open cover of $$\mathsf{supp}\omega$$ by domains of positively or negatively oriented smooth charts, and let $$\{\psi_i\}$$ be a subordinate smooth partition of unity. Define the integral of $$\omega$$ over $$M$$ to be $$\int_M \omega=\sum_i\int_M \psi_i \omega.\tag{16.2}$$

I don't know exactly what guarantees the existence of $$\{U_i\}$$. More concretely, I haven't found any proposition in the previous sections that assures me of a smooth atlas $$\{(U_\alpha,\varphi_\alpha)\}$$ consisting of charts that are either positively-oriented or negatively-oriented. If such atlas existed, I would employ compactness of $$\mathsf{supp}\omega$$ to extract a finite subcover from the open cover $$\{U_\alpha\}$$. This subcover can then serve as $$\{U_i\}$$.

Does anyone have an idea? Thank you.

One way of seeing that this is true is by noting that on an oriented manifold $$(M,\mu)$$, every chart with connected domain is either positively or negatively oriented. To see this, note that in any chart $$\varphi:U\to\mathbb{R}^n$$ we have $$\mu|_{U}=fdx^1\wedge\cdots\wedge dx^n$$, where $$f:U\to\mathbb{R}$$ is continuous and nonvanishing.

From there you can either construct such a covering explicitly (e.g. out of coordinate balls), or note that given any cover by coordinate charts, we can subdivide the disconnected charts into their connected components and obtain a cover by connected coordinate charts.

• I'm not sure if I misunderstood your argument. Are you suggesting that every smooth manifold admits a smooth atlas consisting of smooth charts with connected domain? Thank you.
– Boar
Feb 22, 2022 at 15:10
• @Steve That's also true, but not strictly necessary here. To make sense of the definition in the question, all that's needed is to argue that $\operatorname{supp}(\omega)$ is covered by coordinate charts with connected domain. This can be done e.g. by taking a coordinate ball around each point in $\operatorname{supp}(\omega)$. Feb 22, 2022 at 18:49

Take a look at Proposition 15.6 in ISM:

Proposition 15.6 (The Orientation Determined by a Coordinate Atlas). Let $$M$$ be a smooth positive-dimensional manifold with or without boundary. Given any consistently oriented smooth atlas for $$M$$; there is a unique orientation for $$M$$ with the property that each chart in the given atlas is positively oriented. Conversely, if $$M$$ is oriented and either $$\partial M=\varnothing$$ or $$\dim M >1$$, then the collection of all oriented smooth charts is a consistently oriented atlas for $$M$$.

The existence of a smooth atlas consisting of charts whose coordinate changes are positively (resp. negatively) oriented is equivalent to $$M$$ being orientable, which is a hypothesis we do have in the definition you cite. Since such an atlas does exist (again, because we're supposing $$M$$ is oriented), we can proceed as you say and use the compactness of $$\operatorname{supp}(\omega)$$ to extract the desired finite subcover.

• Thank you, but in Lee's book, a manifold is called orientable if we can endow it with a continuous pointwise orientation, whose definition does not refer to any positively-oriented charts or negatively-oriented charts. That's why I feel confused.
– Boar
Feb 20, 2022 at 2:49