# Differential inclusions for distributions

Given a set valued function $F$ such that for every $x\in M$ (a manifold) we have that $F(x)\subset T_xM$, a differential inclusion is the "equation", $\dot{x} \in F(x)$.

I was wondering if someone has studied an analog of this for distributions, that is, consider a function such that $F(x)\subset (T_xM)^k$ and search for foliations such that in every point are tangent to the span of an element of $F(x)$.

I've tried googleing some similar names and could not find anything, but maybe here someone knows the key-word I am looking for. Also if there is any reference that would be good.

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I don't know about in general, and I suspect this is not exactly what you are looking for, but a very similar problem is used for the method of characteristics for fully nonlinear first order partial differential equations. See en.wikipedia.org/wiki/Monge_cone – Willie Wong Sep 2 '10 at 23:34
Thanks, it was useful, though not exactly what I am looking for. – rpotrie Sep 3 '10 at 14:43
Could you be more specific and tell us where you are heading with this? Could you give an example of what you are aiming for? By `distribution' do you mean (a) a la Schwartz (eg. the Dirac delta function,etc) or (b) a subbundle of the tangent bundle? And by $(T_x M)^k$ do you want symmetric powers $Symm^k (T_x M)$ or the full k-fold tensor product? If you mean $\Lambda^k (T_x M)$ then there is a literature, eg. EDS by Bryant, Chern, et al (available free at the MSRI website) – Richard Montgomery Sep 10 '10 at 3:26
Thanks for the references, I will look at them. By distribution, I mean a subbundle of the tangent bundle, I was wondering under which conditions one can integrate a distribution close to a given one. In mathoverflow.net/questions/37130/… I asked a similar question (based on my ignorance on the subject) and got an answer on some posible restrictions. I was wondering if someone had made a systematic study. – rpotrie Sep 10 '10 at 6:42