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Can we cover the Möbius band with a finite atlas such that the determinant of the Jacobian of each transition map is negative everyhwere?

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    $\begingroup$ It seems doable with 3 charts. $\endgroup$ Oct 19 '11 at 2:26
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To expand on Olivier's comment, you can cover the Möbius band with three rectangles which overlap in thin strips. You can arrange that the transition maps are basically reflection through the $x$-axis, which is a map with negative determinant. One way to visualize this construction is as a band with three half-twists, which is indeed diffeomorphic to the usual Möbius band.

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