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mathphysicist
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A discussion of the kind you want indeed seems to be difficult to locate. I am not an expert but I guess one could prove your claim (and more broadly, some version of the Stokes theorem) for non-orientable manifolds by passing to the two-sheeted covering oriented manifold, as suggested, in a slightly different setting, in the book Geometry VI: Riemannian geometry by Postnikov (here is the link to the relevant page on Google preview).

EDIT: As explained in the Bott--Tu book (see the link in Will's answer and also these two pages), rescuing the Stokes theorem in the non-orientable case requires passing from differential forms to densities.

A discussion of the kind you want indeed seems to be difficult to locate. I am not an expert but I guess one could prove your claim (and more broadly, some version of the Stokes theorem) for non-orientable manifolds by passing to the two-sheeted covering oriented manifold, as suggested, in a slightly different setting, in the book Geometry VI: Riemannian geometry by Postnikov (here is the link to the relevant page on Google preview).

A discussion of the kind you want indeed seems to be difficult to locate. I am not an expert but I guess one could prove your claim (and more broadly, some version of the Stokes theorem) for non-orientable manifolds by passing to the two-sheeted covering oriented manifold, as suggested, in a slightly different setting, in the book Geometry VI: Riemannian geometry by Postnikov (here is the link to the relevant page on Google preview).

EDIT: As explained in the Bott--Tu book (see the link in Will's answer and also these two pages), rescuing the Stokes theorem in the non-orientable case requires passing from differential forms to densities.

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mathphysicist
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A discussion of the kind you want indeed seems to be difficult to locate. I am not an expert but I guess one could prove your claim (and more broadly, some version of the Stokes theorem) for the non-orientable manifolds by passing to the two-sheeted covering orientedoriented manifold, as suggested, in a slightly different setting, in the book Geometry VI: Riemannian geometry by Postnikov (here is the link to the relevant page on Google preview).

A discussion of the kind you want indeed seems to be difficult to locate. I am not an expert but I guess one could prove your claim (and more broadly, some version of the Stokes theorem) for the non-orientable manifolds by passing to the two-sheeted covering oriented manifold, as suggested, in a slightly different setting, in the book Geometry VI: Riemannian geometry by Postnikov (here is the link to the relevant page on Google preview).

A discussion of the kind you want indeed seems to be difficult to locate. I am not an expert but I guess one could prove your claim (and more broadly, some version of the Stokes theorem) for non-orientable manifolds by passing to the two-sheeted covering oriented manifold, as suggested, in a slightly different setting, in the book Geometry VI: Riemannian geometry by Postnikov (here is the link to the relevant page on Google preview).

Source Link
mathphysicist
  • 5.3k
  • 1
  • 43
  • 44

A discussion of the kind you want indeed seems to be difficult to locate. I am not an expert but I guess one could prove your claim (and more broadly, some version of the Stokes theorem) for the non-orientable manifolds by passing to the two-sheeted covering oriented manifold, as suggested, in a slightly different setting, in the book Geometry VI: Riemannian geometry by Postnikov (here is the link to the relevant page on Google preview).