Structure results for the integral set of a submanifold with respect to a non-integrable exterior differential system.

Let N and O be, respectively, a C 2 manifold and an arbitrary family of C 1 differential forms on N . Moreover, assume that For all y ∈ N and for all M-dimensional integral elements Σ of O at y, there is ω ∈ O such that (dω)y|Σ ̸= 0. (∗) If M is any M-dimensional C 1 imbedded submanifold of N , then we expect that condition (∗) prevents the existence of interior points in the integral subset of M with respect to O, i.e., I(M, O) := \ ω∈O {ω|M = 0}. Actually, the structure of I(M, O) can be described much more precisely by invoking the notion of superdensity. Indeed, under the previous hypotheses, the following structure result holds: There are no (M + 1)-density points of I(M, O) relative to M. If we now consider M in the smaller class of C 2 imbedded submanifolds of N , then it becomes natural to expect a further “slimming” of I(M, O). Indeed, we have the following second structure result: If O is countable, then I(M, O) is an (M − 1)-dimensional C 1 rectifiable subset of M. These results are immediate corollaries of two general structure theorems, which are the main goal of this paper. Applications in the context of noninvolutive distributions and in the context of the Pfaff problem are provided.