A smooth manifold $M$ is compact and oriented: it carries a compact topology together with an (abstractly recorded) orientation, the standard hypothesis under which Stokes-type integration is defined.
- compact : CompactSpace M
- oriented : True
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Projects out the underlying CompactSpace structure of a compact oriented manifold.
For a compact oriented smooth manifold $(M, I)$, this produces the Stokes-style integration data on its differential form space: the integration map $\int_M : \Omega^{\dim M}(M) \to \mathbb{R}$ and associated pairings used to formulate Stokes' theorem.
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The dimension $\dim M$ of the manifold $M$, as recorded by its Stokes integration data.
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The top-degree integration map $\int_M : \Omega^{\dim M}(M) \to \mathbb{R}$ on a compact oriented smooth manifold.
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The period pairing $\Omega^p(M) \times \Omega^p(M) \to \mathbb{R}$ induced by integration on $M$, generalising $\langle \alpha, \beta \rangle = \int_M \alpha \wedge \star \beta$.