Briefly, my question is, what algorithms or techniques exist for determining the maximal dimension of an integral submanifold of a non-integrable distribution?
The Heisenberg Group $ \mathbb{H}_n $ is the "simplest" example of a subriemannian manifold. It can be presented as $ \mathbb{R}^{2n+1} $ (with coordinates $ (x_1, y_1, \ldots, x_n, y_n, t) $) equipped with the horizontal distribution $ H \mathbb{H}_n = \ker(\alpha) $ where $ \alpha $ is the one-form $$ \alpha = dt + \sum_{i=1}^{n}{x_i\, dy_i} $$ One can show that, for $ k \geq n+1 $, every (say, smooth) $ k $-form $ \kappa $ can be decomposed as \begin{equation}\label{decomp}\tag{1} \kappa = \alpha \wedge \sigma + d \alpha \wedge \tau \end{equation} for suitable (smooth) forms $ \sigma $ and $ \tau $. It therefore follows that there do not exist smooth horizontal embeddings $ f : \mathbb{R}^{k} \rightarrow \mathbb{H}_n $ for such $ k $. By horizontal, we mean, $ f^* \alpha = 0 $. Indeed, if $ f : \mathbb{R}^k \rightarrow \mathbb{H}_n $ is any smooth map, then \eqref{decomp} implies $ f^* \kappa = 0 $ for any $ k $-form $ \kappa $, whence $ f $ has rank $ \text{rank}{(Df)} < k $. This example shows that not only is the $ 2n $-dimensional horitontal distribution $ H \mathbb{H} $ not integrable, but in fact, there are not even $ (n+1) $-dimensional horizontal submanifolds (!)
Proving this fact was not easy, essentially because it was not very easy to prove \eqref{decomp}. All I found was a long, elementary combinatorial argument. I am aware of the fact that an $m$-dimensional distribution is integrable - i.e., has $ m $-dimensional integral manifolds - but I am not aware of how one looks for maximal submanifolds of a non-integrable distribution such as $ H \mathbb{H}_n $ above. Some nice pointers I am looking for include:
- An theorem (or better yet, an algorithm) to calculate the maximum dimension of an integral submanifold of a distribution $ HM \subseteq TM $ in the tangent space of a manifold $ M $ (presented as a span of vector fields or as the intersection of kernels of one-forms).
- An example of a distribution distinct from the Heisenberg groups $ \mathbb{H}_n $ exhibiting this non-integrability. In fact, let's introduce the following terminology: let's say that a distribution is $ k $-integrable if it has $ k $-dimensional, but not $ (k+1) $-dimensional, integral submanifolds. For what pairs $ (m,k) $ can we construct an $ m $-dimensional $ k $-integrable distribution on $ \mathbb{R}^n $ ($ n > m $)?
- References in which these or related questions are systematically studied.
Thanks for your attention!
The subject of exterior differential systems was essentially designed to answer exactly these sorts of questions, and much of the theory is algorithmic.
I don't have time to give detailed thought to your question, I'm afraid, but can point you to two standard references on the subject:
(1) Nine Lectures on Exterior Differential Systems by Bryant.
(2) Exterior Differential Systems by Bryant, Chern, Gardner, Goldschmidt, and Griffiths.
Chapter II of reference (2) is especially relevant to your question. Key words you might want in searching for more literature: The Pfaff Theorem (or Pfaff-Darboux Theorem), Pfaffian Systems, Cartan-Kahler Theory.