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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differential-geometry
integration
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4I don't remember the details, but there is a notion of integration on nonorientable manifolds. You could always integrate on the orientation cover, but you can also define modified top dimensional differential forms by setting $|\Lambda^n(M)|$ the orientable real line bundle obtained from the cocycle $|\det(Jac)|$ the determinant of the jacobian matrices associated to a chart, you can even define $|\Lambda^n(M)|^p$ for $1\leq p<\infty$ with the cocycle $|\det(Jac)|^p$. There is then a natural way to "integrate the $p$-th power of sections of this bundle. – 2012-11-06
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2Or you could remove a measure-zero subset making $M$ orientable. – 2012-11-06