A characterization of Pfaffian embeddings from (2, 3, 5)- into flat (4, 7)-geometries

TL;DR

This paper investigates the conditions under which (2,3,5)-manifolds can be Pfaffian embedded into flat (4,7)-geometries, revealing that only certain manifolds with specific curvature properties are embeddable.

Contribution

It provides a complete characterization of embeddable (2,3,5)-manifolds based on their Cartan geometries and curvature conditions, addressing a specific case of Pfaffian embeddings.

Findings

Generic (2,3,5)-manifolds do not embed due to curvature obstructions.

Embeddability requires a double root in the Cartan quartic at each point.

Characterization depends on higher order curvature invariants.

Abstract

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.