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

TL;DR

This paper explores the geometric properties of rank-3 vector bundles over surfaces, focusing on 2-plane distributions in 5-manifolds, and links surface geometry in projective space to these distributions.

Contribution

It establishes a correspondence between surface projective geometry and 2-plane distributions in 5-dimensional bundles, revealing new geometric insights.

Findings

Any surface in 3D projective space can be associated with such a 5D geometric structure.

A dictionary is established linking projective differential geometry to the growth vector of distributions.

The work provides a new perspective on the interplay between surface geometry and higher-dimensional distributions.

Abstract

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.