Nearly Kähler geometry and $(2, 3, 5)$-distributions via projective holonomy

Published in Indiana University Mathematics Journal, 2017, with A.R. Gover and R. Panai
(iumj.indiana.edu | arXiv:1403.1959)

Abstract

We show that any dimension-$6$ nearly Kähler (or nearly para-Kähler) geometry arises as a projective manifold equipped with a $G_2^{(∗)}$ holonomy reduction. In the converse direction, we show that if a projective manifold is equipped with a parallel seven-dimensional cross product on its standard tractor bundle, then the manifold is a Riemannian nearly Kähler manifold, if the cross product is definite; otherwise, if the cross product has the other algebraic type, the manifold is in general stratified with nearly Kähler and nearly para-Kähler parts separated by a hypersurface that canonically carries a Cartan $(2, 3 ,5)$-distribution. This hypersurface is a projective infinity for the pseudo-Riemannian geometry elsewhere on the manifold, and we establish how the Cartan distribution can be understood explicitly and also (in terms of conformal geometry) as a limit of the ambient nearly (para-)Kähler structures. Any real-analytic $(2, 3, 5)$-distribution is seen to arise as such a limit, because we can solve the geometric Dirichlet problem of building a collar structure equipped with the required holonomy-reduced projective structure. A model geometry for these structures is provided by the projectivization of the imaginary (split) octonions. Our approach is to use Cartan/tractor theory to provide a curved version of this geometry; this encodes a curved version of the algebra of imaginary (split) octonions as a flat structure over its projectivization. The perspective is used to establish detailed results concerning the projective compactification of nearly (para-)Kähler manifolds, including how the almost (para-)complex structure and metric smoothly degenerate along the singular hypersurface to give the distribution there.