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Paraconformal geometry of $n$th order ODEs, and exotic holonomy in dimension four | | Maciej Dunajski
; Paul Tod
; | | Date: |
24 Feb 2005 | | Abstract: | 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 $ngeq 6$ 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. | | Source: | arXiv, math.DG/0502524 | | Services: | Forum | Review | PDF | Favorites | |
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