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    • CommentRowNumber1.
    • CommentAuthorzskoda
    • CommentTimeNov 30th 2011
    • (edited Nov 30th 2011)

    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 UU be an open set on a smooth Hausdorff paracompact manifold MM of dimension m+nm+n. The following is equivalent for a distribution HH of subspaces in the tangent bundle TMTM: (i) There are nn smooth forms on UU such that H pH_p is the common annihilator of them at every point pUp\in U; (ii) there exist mm smooth vector fields such that H pH_p is the span of those at every point pUp\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 UU. But I do not see that the trivializing mm-tuple on the form side would imply global trivializing nn-tuple on the vector field side for general open set UMU\subset M.

    Any help ?