[Math] Different definitions of closed differential form

differential-formsdifferential-geometryintegration

I have recently read, out of curiosity, Tao's very motivating introduction to Differential forms and integration. At pages 5 and 8 he seems to define a closed $k-$form (on some manifold) to be a $k-$form whose integral along any closed $k-$submanifold (i.e., one with zero boundary) vanishes. I am used, instead, to the definition that a form is closed iff its differential is zero. I think that in general Tao's definition is strictly stronger than mine. Here are my thoughts. In the following, we only consider oriented (and, if necessary, compact) manifolds. I believe that the following conditions should all be equivalent for a $k-$form $\omega$:

  1. $d\omega=0$ (my definition of being closed)

  2. $\int_S d\omega = 0$ for all $(k+1)-$submanifolds $S$

  3. $\int_{\partial S} \omega=0$ for all $(k+1)-$submanifolds $S$, i.e., $\omega$ vanishes on all exact submanifolds

The equivalence of 2. and 3. is given by Stokes' theorem. 1. clearly implies 2. and I think that 2. implies 1. because the integration pairing is non-degenerate.

In particular, if the above holds, I think that Tao's condition is equivalent to mine iff the $k-$th singular homology of the ambient manifold vanishes $(*)$.

I thought maybe de Rham's theorem could solve the problem, but if I interpret it right, it only implies that $\omega$ is exact if and only if its integration along every closed $k-$submanifold gives $0$ $(!)$. Which is kind of dual to 1. $\Leftrightarrow$ 3. above.

Now, it seems very strange to me that Tao should use a definition that is not equivalent to the usual one, so I think I must be missing something. Please help me to answer the following:

  • A) Are my three conditions above equivalent and is $(!)$ correct?
  • B) Is $(*)$ right? In particular, when considering singular homology of manifolds, can we restrict to smooth submanifolds (with boundary) instead of arbitrary continuous images of simplices?
  • C) Did I get Tao's definition of closed form right? If so, is it really different from mine? Why?

Best Answer

I will post this as an answer, since there are at least 4 people interested to know how this ends: I have informed professor Tao of the possible error and he has immediately acknowledged it and told me that he has corrected it. The link to the notes does not seem to be working at present, but I think it is probably just a temporary disruption.

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