A form $\beta^{p}$ is closed if $d\beta=0$.

A form $\beta^{p}$ is exact if $\beta^{p}=d\alpha^{p-1}$, for some form $\alpha^{p-1}$.

An orientable closed manifold is a compact manifold without boundary.

What is a compact manifold?

Why is an orientable closed manifold the same as a compact manifold without boundary?

If a manifold is compact, does it not, be definition, have a boundary?

How do you prove that the integral of an exact form over an orientable closed manifold is $0$?


closed as too broad by Andrew D. Hwang, Willie Wong, Jack Lee, pjs36, MickG Nov 10 '16 at 20:07

Please edit the question to limit it to a specific problem with enough detail to identify an adequate answer. Avoid asking multiple distinct questions at once. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.

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    $\begingroup$ Voting to close as "missing context": If you don't know what a compact manifold (such as a circle, a sphere, a torus...) is, explaining integration of forms is the subject of a short course or textbook, not a Math.SE post. $\endgroup$ – Andrew D. Hwang Nov 10 '16 at 11:40

I will try to answer your questions:

1) A manifold is compact if every open covering (i.e. a collection of open sets that contains the manifold in its union) has a finite subcovering. So from this open covering, which might have infinitely many sets, you can choose finitely many and these must still cover the manifold.

2)On this website https://en.wikipedia.org/wiki/Classification_of_manifolds a closed manifold is defined as a compact manifold without boundary.

3) No, a compact manifold could have no boundary. For example, consider the circle $S^1$. It is definitely compact and does not have a boundary.

4) As we have just concluded that this manifold has no boundary, we can just use Stoke's theorem for our manifold $M$: \begin{equation} \int_{M} d\omega = \int_{\partial M} \omega. \end{equation} We have an exact form, let's call it $\beta^p$. As you already said, this means that $\beta^p=d\alpha^{p-1}$. Also, because $M$ has no boundary, the integral over $\partial M$ will be zero: \begin{equation} \int_M \beta^p = \int_M d\alpha^{p-1} = \int_{\partial M} \alpha^{p-1} =0. \end{equation}

  • $\begingroup$ What is an example of a compact manifold with boundary? $\endgroup$ – nightmarish Nov 10 '16 at 11:50
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    $\begingroup$ Just an interval for example: [-1,1] $\endgroup$ – Hugo Berndsen Nov 10 '16 at 13:07

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