Geometric aspects of rank-3 vector bundles over surfaces and 2-plane distributions on 5-manifolds

We study geometric aspects of horizontal 2-plane distributions on the complement of the zero section in the 5-dimensional total space of a rank-3 vector bundle equipped with connection over a surface. We show that any surface in 3-dimensional projective space can be associated to such a geometric structure in 5-dimensions, and establish a dictionary between the projective differential geometry of the surface and the growth vector of the 2-plane distribution.

💡 Research Summary

The paper establishes a novel bridge between the classical projective differential geometry of surfaces in three‑dimensional projective space and the modern theory of rank‑2 distributions on five‑dimensional manifolds. Starting with a smooth surface Σ (real or complex) immersed in ℙ³, the authors consider the pull‑back of the ambient tangent bundle, which yields a rank‑3 vector bundle B → Σ equipped with a natural GL(3)‑connection ∇ that combines the Levi‑Civita connection on the tangent bundle and the normal connection determined by the second fundamental form.

By removing the zero section Z_B from the total space Tot(B) they obtain a 5‑manifold M⁵ = Tot(B) \ Z_B. Pulling back the connection 1‑form ω to M⁵ produces a rank‑3 exterior differential system (EDS) Ω = ⟨π*ω⟩. The annihilator of Ω defines a rank‑2 distribution Δ ⊂ TM⁵, explicitly generated by two vector fields X and Y (formula (1.6) in the paper). This distribution is precisely the horizontal lift of the tangent bundle of Σ, while the vertical directions correspond to the fibers of B.

The authors then study the derived flag of Δ: Δ ⊂ Δ′ = Δ +