Conformal Geometry and 3-plane Fields on 6-manifolds

The purpose of this note is to provide yet another example of the link between certain conformal geometries and ordinary differential equations, along the lines of the examples discussed by Nurowski [3]. In this particular case, I consider the equivalence problem for 3-plane fields D ⊂ TM on a 6-manifold M satisfying the nondegeneracy condition that D + [D, D] = TM . I give a solution of the equivalence problem for such D (as Tanaka has previously), showing that it defines a so(4, 3)-valued Cartan connection on a principal right H-bundle over M where H ⊂ SO(4, 3) is the subgroup that stabilizes a null 3-plane in R. Along the way, I observe that there is associated to each such D a canonical conformal structure of split type on M , one that depends on two derivatives of the plane field D. I show how the primary curvature tensor of the Cartan connection associated to the equivalence problem for D can be interpreted as the Weyl curvature of the associated conformal structure and, moreover, show that the split conformal structures in dimension 6 that arise in this fashion are exactly the ones whose so(4, 4)-valued Cartan connection admits a reduction to a spin(4, 3)connection. I also discuss how this case is analogous to features of Nurowski’s examples.