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Sep 29, 2011 at 13:26 comment added Willie Wong I'm accepting this answer because, as pointed out by Michael, the issue is mostly topological. Deane's comments also helped greatly. In particular, Goldschmidt's 1967 JDG paper "Integrability Criteria..." mentioned the same issue, namely that the topological condition from the point of view of a submersion maybe too restrictive (see the paragraph before Defn. 7.2).
Sep 29, 2011 at 13:23 vote accept Willie Wong
Sep 28, 2011 at 16:16 history edited Michael Bächtold CC BY-SA 3.0
added 1120 characters in body; added 72 characters in body
Sep 28, 2011 at 12:37 comment added Willie Wong Of course. I wasn't clear about it in my question (sorry!), but of particular interest would also be the issue of when $H$ can be taken to be a submersion. In some sense the submersion requirement would be a good one, since for "dimension counting" purposes (whether an equation is over/under/just-okay-determined) it is usually a good idea for the equation to be defined by $\mathcal{R}$ a submanifold, not just an arbitrary closed set.
Sep 28, 2011 at 10:31 history answered Michael Bächtold CC BY-SA 3.0