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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?

Michael Hardy
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Joel J.
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1 Answers1

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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.

Cheerful Parsnip
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