Is every orientation form on a compact smooth manifold closed?

[robforsub]

Is every orientation form w on a compact smooth manifold closed?(i.e. dw=0)

 

[Bacle]

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.

 

[quasar987]

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.