Structure results for the integral set of a submanifold with respect to a non-integrable exterior differential system
Volume 131 / 2023
Abstract
Let $\mathcal {N}$ and $\mathcal {O}$ be, respectively, a $C^2$ manifold and an arbitrary family of $C^1$ differential forms on $\mathcal N $. Moreover, assume that $$\begin{aligned} &\llap{(\ast)}\ \text{For all $y\in \mathcal {N}$ and for all $M$-dimensional integral elements}\\ &\ \text {$\Sigma $ of $\mathcal {O}$ at $y$, there is $\omega \in \mathcal {O}$ such that $(d\omega )_y\vert _\Sigma \not =0$.} \end{aligned}$$ If $\mathcal {M}$ is any $M$-dimensional $C^1$ imbedded submanifold of $\mathcal {N}$, then we expect that condition $(*)$ prevents the existence of interior points in the integral subset of $\mathcal {M}$ with respect to $\mathcal {O}$, i.e., $$ \mathcal I (\mathcal {M},\mathcal {O}):=\bigcap _{\omega \in \mathcal {O}} \{\omega \vert _{\mathcal M} =0\}. $$ Actually, the structure of $\mathcal I (\mathcal {M},\mathcal {O})$ 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 $\mathcal I (\mathcal {M},\mathcal {O})$ relative to $\mathcal {M}$.
If we now consider $\mathcal {M}$ in the smaller class of $C^2$ imbedded submanifolds of $\mathcal N $, then it becomes natural to expect a further “slimming” of $\mathcal I (\mathcal {M},\mathcal {O})$. Indeed, we have the following second structure result: If $\mathcal {O}$ is countable, then $\mathcal I (\mathcal {M},\mathcal {O})$ is an $(M-1)$-dimensional $C^1$ rectifiable subset of $\mathcal {M}$. These results are immediate corollaries of two general structure theorems, which are the main goal of this paper. Applications in the context of non-involutive distributions and in the context of the Pfaff problem are provided.
