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1. • CommentRowNumber1.
   • CommentAuthorzskoda
   • CommentTimeNov 30th 2011
   • (edited Nov 30th 2011)
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   Author: zskoda
   Format: MarkdownItex[[M M Postnikov]]'s books on geometry and topology are among my personal favourites. Careful teahcing with love and elegance, precision in theory and with lots of examples elaborated in great detail. It is also very reliable. I have however problem with one statement which I found few times in his books and which I have problem with: > Let $U$ be an open set on a smooth Hausdorff paracompact manifold $M$ of dimension $m+n$. The following is equivalent for a distribution $H$ of subspaces in the tangent bundle $TM$: (i) There are $n$ smooth forms on $U$ such that $H_p$ is the common annihilator of them at every point $p\in U$; (ii) there exist $m$ smooth vector fields such that $H_p$ is the span of those at every point $p\in U$. So, I have no problem in proving that this is true locally, or I think it holds more strongly, over a contractible open $U$. But I do not see that the trivializing $m$-tuple on the form side would imply global trivializing $n$-tuple on the vector field side for general open set $U\subset M$. Any help ?

   M M Postnikov’s books on geometry and topology are among my personal favourites. Careful teahcing with love and elegance, precision in theory and with lots of examples elaborated in great detail. It is also very reliable. I have however problem with one statement which I found few times in his books and which I have problem with:

   Let $U$ be an open set on a smooth Hausdorff paracompact manifold $M$ of dimension $m+n$. The following is equivalent for a distribution $H$ of subspaces in the tangent bundle $TM$: (i) There are $n$ smooth forms on $U$ such that $H_p$ is the common annihilator of them at every point $p\in U$; (ii) there exist $m$ smooth vector fields such that $H_p$ is the span of those at every point $p\in U$.

   So, I have no problem in proving that this is true locally, or I think it holds more strongly, over a contractible open $U$. But I do not see that the trivializing $m$-tuple on the form side would imply global trivializing $n$-tuple on the vector field side for general open set $U\subset M$.

   Any help ?