
6. Miscellaneous

1. Problem 6.1.

If two smooth manifolds $M_1$ and $M_2$ are homeomorphic, could their spaces of Engel structures be used to distinguish them smoothly?
1. Remark. One could try to connect Engel structures to gauge theoretic invariants, or to symplectic structures. The latter are known to distinguish certain smooth structures.
• Problem 6.2.

Are there examples of pairs $(n,k)$ with $n>k\ge2$ such that for any parallelizable $n$ manifold, there exists a $k$-plane field $\mathcal{D}\subset TM$ with maximal growth vector?
1. Remark. More generally, instead of parallelizability, one would like to assume only that one is dealing with manifolds admitting appropriate partial flags.

Cite this as: AimPL: Engel structures, available at http://aimpl.org/engelstr.