I feel it rather weird that there is a notion of integration when you glue a patch of paper to get a surface of cylinder while there is not a suitable notion when you glue it differently to get a Moebius band. Should not a Moebius band have a definite area??
why we cannot integrate on a nonorientable manifold?
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$\begingroup$
differential-geometry
integration
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2Or you could remove a measure-zero subset making $M$ orientable. – 2012-11-06