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Forums
Mathematics
Differential Geometry
Fixing an orientation for a connected smooth surface
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[QUOTE="fresh_42, post: 6382149, member: 572553"] What do you want to prove, the Lemma that there are only two equivalence classes, or that it is sufficient to consider the first chart of an oriented atlas to determine the orientation? It would be convenient to consider the problem in the language of differential forms to prove the latter. Do you have a definition of oriented manifolds in terms of differential forms, or just that the charts are equally oriented as mentioned above? And do we have a compact, connected manifold? The idea is in both cases that we consider a path of charts from ##x_0## to ##x## (existence by connectivity, finiteness by compactness). Then all matrices of the basis changes from one chart to the next have a positive determinant. Multiplying them along the path doesn't change the sign, so the orientation of the first chart is all we need to know. [/QUOTE]
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Differential Geometry
Fixing an orientation for a connected smooth surface
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