Is every orientation form on a compact smooth manifold closed?

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SUMMARY

Every orientation form on a compact smooth manifold is closed, meaning that the exterior derivative of the orientation form, denoted as dw, equals zero. This conclusion is based on the algebraic property that the (n+k)-th exterior power of an n-dimensional vector space is zero for any k greater than zero. Therefore, since the exterior derivative of an n-form results in an (n+1)-form, it follows that the orientation form on an n-manifold is indeed closed.

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Is every orientation form w on a compact smooth manifold closed?(i.e. dw=0)
 
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Unless I am missing something, yes, since the exterior derivative of an n-form would
be (is) an (n+1)-form in an n-manifold.
 
Correct. It is an algebraic fact that the (n+k)-th exterior power of an n-dimensional vector space is 0 for any k>0. Since the exterior derivative of an orientation form on an n-manifold is an object of the (n+1)-th exterior power of the cotangent bundle, it is 0.
 

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