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Nov 30, 2016 at 9:57 vote accept Ali Taghavi
Nov 30, 2016 at 9:02 comment added Andreas Cap I have added more details to the answer.
Nov 30, 2016 at 9:02 history edited Andreas Cap CC BY-SA 3.0
added more detail
Nov 29, 2016 at 21:22 comment added Ali Taghavi May I am missing some thing but assume that B1 and B2 are two forms on M which agree on the boundary this does not imply that dB1 and dB2 agree on the boundary. So if I am not mistaken, some thing is missing in your answer.
Nov 29, 2016 at 9:20 comment added Andreas Cap Any form on the boundary extends to a form on $M$, and the exterior derivative commutes with restriction to the boundary (which is a pullback). Hence any exact form on the boundary is the restriction of an exact form on $M$.
Nov 29, 2016 at 8:34 comment added Ali Taghavi but your argument shows that $I_{2}$ is zero on the restriction of every exact form to the boundary. But this does not imply that $I_{21}$ vanishs at exact forms on the boundary which are not necessarily restriction of an exact form from $M$ to $\partial M$? Am I mistaken?
Nov 29, 2016 at 8:20 vote accept Ali Taghavi
Nov 29, 2016 at 8:34
Nov 29, 2016 at 8:11 history answered Andreas Cap CC BY-SA 3.0