Paraconformal geometry of $n$th order ODEs, and exotic holonomy in dimension four

英文摘要：

We characterise $n$th order ODEs for which the space of solutions $M$ is equipped with a particular paraconformal structure in the sense of \cite{BE}, that is a splitting of the tangent bundle as a symmetric tensor product of rank-two vector bundles. This leads to the vanishing of $(n-2)$ quantities constructed from of the ODE. If $n=4$ the paraconformal structure is shown to be equivalent to the exotic ${\cal G}_3$ holonomy of Bryant. If $n=4$, or $n\geq 6$ and $M$ admits a torsion--free connection compatible with the paraconformal structure then the ODE is trivialisable by point or contact transformations respectively. If $n=2$ or 3 $M$ admits an affine paraconformal connection with no torsion. In these cases additional constraints can be imposed on the ODE so that $M$ admits a projective structure if $n=2$, or an Einstein--Weyl structure if $n=3$. The third order ODE can in this case be reconstructed from the Einstein--Weyl data.