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How can I show that if an atlas on a smooth manifold has exactly 2 charts then it is orientable? How do I make sure that the Jacobian of the transition map is positive?

bluebox
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    The Mobius band has an atlas with exactly 2 charts. – Lee Mosher Apr 16 '14 at 15:44
  • Yes, you are absolutely right. How do I delete my question? – bluebox Apr 16 '14 at 16:06
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    This good question shouldn't be deleted. It is indeed quite plausible (but false) that an atlas with two charts can be oriented. A good answer should show why the obvious "proof" does not work and what purely toplogical supplementary hypothesis on the two domains of the charts ensures that the result is true. – Georges Elencwajg Apr 16 '14 at 17:31
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    Here's a modification of your statement that is true: If $M$ is a smooth manifold that admits an atlas with exactly two charts whose intersection is connected, then $M$ is orientable. Can you prove that? – Jack Lee Apr 17 '14 at 17:30
  • I will think about this Prof. Lee – bluebox Apr 17 '14 at 17:47

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