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On pg. 417 of Lee's Introduction to Smooth Manifolds, the author discusses integration on a manifold with corners.

The text states:

... for Stokes's theorem, we need to integrate a differential form over a smooth manifold with corners. Since the boundary (of a smooth manifold with corners) is not itself a smooth manifold with corners, this requires a separate definition. Let $M$ be an oriented smooth $n$-manifold with corners, and suppose $\omega$ is an $(n-1)$-form on $\partial M$...

I am perplexed. How can we talk about an $(n-1)$-form on $\partial M$ when $\partial M$ may not even be a smooth manifold? Am I missing something obvious?

Thanks.

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  • $\begingroup$ It's just a section of the restricted bundle $\Lambda^{n-1}T^*M|_{\partial M}$, that is, a continuous map $\omega\colon \partial M \to \Lambda^{n-1}T^*M$ such that $\omega_p\in \Lambda^{n-1}(T^*_pM)$ for each $p$. (I remarked in the paragraph just before the one you quoted that the usual flora and fauna of smooth manifolds, including differential forms, can be defined on smooth manifolds with corners in the same way as for smooth manifolds and smooth manifolds with boundary.) $\endgroup$ – Jack Lee Feb 26 '17 at 1:10
  • $\begingroup$ OK, I've done that. $\endgroup$ – Jack Lee Feb 26 '17 at 6:33
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An $(n-1)$ form on $\partial M$ is just a section of the restricted bundle $\Lambda^{n-1}T^*M|_{\partial M}$, that is, a continuous map $\omega\colon \partial M\to \Lambda^{n-1}T^*M$ such that $\omega_p\in \Lambda^{n-1}(T_p^*M)$ for each $p$. (I remarked in the paragraph just before the one you quoted that the usual flora and fauna of smooth manifolds, including differential forms, can be defined on smooth manifolds with corners in the same way as for smooth manifolds and smooth manifolds with boundary.)

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