The aim of the paper under review is to show clearly the link between projective differential geometry and nearly Kähler geometry and, using this, to study a convergence of nearly Kähler geometry and \((2,3,5)\)-geometry.
The authors prove that any 6-dimensional nearly Kähler (or nearly para-Kähler) manifold is a projective manifold endowed with a \(G_2^{*}\) holonomy reduction. Conversely, it is shown that if a projective manifold is equipped with a parallel 7-dimensional cross product on its standard tractor bundle, then the manifold is a Riemannian nearly Kähler manifold, if the cross product is definite; otherwise, the manifold is stratified by nearly Kähler and nearly para-Kähler parts separated by a hypersurface which carries a canonical Cartan \((2,3,5)\)-distribution.
For proving this, a geometric Dirichlet problem is solved.
A model geometry for these structures is provided by the projectivization of the imaginary octonions.
Interesting supporting examples are given.