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I am reading Guillemin & Pollack's Differential Topology and I love it so far. However, the chapter about orientation of a manifold makes me struggle quite a bit. I have yet fully understood it. In particular, I am lost every time the book says showing some way of defining an orientation on a manifold is smooth. For example, here is an excerpt

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I totally understand the way an orientation is defined on S as the book described. But then I don't know how to show "smoothness" of this orientation.

My thinking is as follows. Please let me know if I am on the right track.

I am supposed to show that, for each $x \in S$, there exists a local parametrization $\phi :U\subset R^k \to E \subset S$, $\phi(0)=x$ such that $d\phi_u$ is an orientation preserving isomorphism with respect to the standard orientation on $R^k$, which is $\{e_1, e_2, ..., e_n\}$ being positive orientation, and the orientation on $T_u(S)$ constructed according to the book, for each $u\in E$.

Then I have to show that for any such two overlapping parametrization, the derivative of transition function has positive determinant.

Do you guys think there is a better way of showing "smoothness" than this. And please do not use anything not in GP's book.

Thank you.

Nick
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  • No, you (they) have already defined an orientation at each point, so there's no need to do overlaps. To check smoothness, all you have to do is verify that your already-defined orientations are consistent with one another on any connected open subset coming from a parametrization. The patching together with overlaps is a different approach, where one starts with an orientation on one parametrization and then tries to define the orientation elsewhere by patching/gluing together. Have you been looking at other books/posts? – Ted Shifrin May 14 '22 at 16:44

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