  Given two smooth manifolds with tangent subbundle distributions, an embedding
is Pfaffian if its differential sends the distribution on the source into the
distribution on the target. In this paper, we consider the question of
existence of Pfaffian embeddings in the specific case where the source is a
(2,3,5)-manifold, the target is the 7-dimensional space of isotropic 2-planes
in a 6-dimensional symplectic vector space, and the Pfaffian condition is that
the derived 3-distribution on the source be mapped into the natural
4-distribution on the target. This is one of the simpler non-trivial cases of
the general question on existence of Pfaffian embeddings, but already the
answer here requires solution of an interesting differential equation. It turns
out that a generic (2,3,5)-manifold does not embed, the first obstruction being
the fact that a Pfaffian embeddable (2,3,5)-manifold necessarily has a double
root for its Cartan quartic at each point. We determine a complete
characterization of embeddable (2,3,5)-manifolds in terms of their associated
Cartan geometries, which characterization depends on higher order
(non-harmonic) curvature as well.
