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Loring Tu-Introduction to manifolds

This is a section from Loring Tu's book Introduction to Manifolds page 244 Second Edition. My question is as follows:

Towards the end of the text in the image he says that an oriented manifold can be described by a pair (M,[w]) where [w] is the equivalence class of an orientation form on M. But a little above he says that such an equivalence class can correspond to an orientation on a "connected" orientable manifold. So if we have an arbitrary orientable manifold which can possibly have many connected components how can this pair be well defined. What I mean to say is different connected components can have different orientations and therefore different equivalence class of smooth nowhere-vanishing n-forms.So how is he able to select "ONE" such equivalence class of an orientation form if the manifold is not connected? Please let me know if I need to add more details or the question itself is not clear.

Thanks.

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  • $\begingroup$ Indeed, you just need define an orientation form in each connected component $\endgroup$ – DiegoMath Apr 18 at 15:57
  • $\begingroup$ @DiegoMath Can you please elaborate.I am very confused at this point.Suppose we define an orientation form in each connected component.Then how does that define an orientation form on M as stated in the image?What am I missing here? $\endgroup$ – RagingBull Apr 18 at 16:00
  • $\begingroup$ also what does this statement mean then: An oriented manifold can be described by a pair (M,[w])? $\endgroup$ – RagingBull Apr 18 at 16:25

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