Generic plane fields and special holonomy

Travis Willse
(Australian National University)

Abstract: Nurowski used Cartan's solution of the local equivalence problem for generic 2-plane fields on 5-manifolds to show that any such plane field induces a canonical signature-(2,3) conformal structure on the underlying manifold. Exploiting a local normal form given by Cartan, we consider a natural infinite-dimensional class of highly symmetric 2-plane fields (which includes the submaximally symmetric plane fields) and show that the conformal structures they induce enjoy additional geometric structures. We produce explicit Fefferman-Graham ambient metrics for these conformal structures and show that the local holonomy of these metrics and the local normal conformal holonomy of the underlying structures are equal to the Heisenberg 5-group; this holonomy group acts indecomposably but not irreducibly.