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I learned about orientations on manifolds and induced orientation on the surface and I don't get the idea. When I search online I find explanations with different definitions.

Here is how we defined orientaion: orientation definition

The induced orientation was defined using compostion of the orientation with the containment function multiplied with $(-1)^m$ where m is the dimension of the manifold.

What is the $(-1)^m$ for? I would be glad to get an intuitive explanation and an example of induced orientation on a simple manifold (preferably one from odd dimension).

Amit Keinan
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    The funny sign ensures that Stokes theorem becomes $\int_Md\omega=\int_{\partial M}\omega$. – peek-a-boo Jul 03 '21 at 17:52
  • You don't say so, but, I presume that $M$ is a manifold with boundary? And that the "induced orientation" is an orientation defined on its boundary $\partial M$? – Lee Mosher Jul 03 '21 at 17:54
  • @LeeMosher yes and yes :) – Amit Keinan Jul 03 '21 at 18:01
  • What does " compostion of the orientation with the containment function " mean? – Arctic Char Jul 03 '21 at 18:10
  • @Arctic Char $Ox0 = (-1)^m * (Ox composition i)$ where $i: R^(m-1) -> H^m i(u1, ..., um-1) = (u1, ..., um-1,0)$ – Amit Keinan Jul 03 '21 at 18:38
  • This answer explain briefly the term $(-1)^m$. A more detailed explanation requires some understanding of differential forms and ultimately the Stokes theorem. If you are not there yet, I suggest you come back to this later. – Arctic Char Jul 03 '21 at 18:45

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