Parallel (co-)tractors and the geometry of first BGG solutions on almost Grassmannian structures

TL;DR

This paper investigates the geometric structures of almost Grassmannian manifolds by explicitly formulating tractor bundles, BGG operators, and analyzing solutions and their geometric implications.

Contribution

It provides explicit formulas for splitting operators, BGG operators, and prolongation connections in almost Grassmann structures, and characterizes solutions geometrically.

Findings

Explicit formulas for splitting operators and BGG operators.

Characterization of parallel tractors and cotractors.

Description of geometry on the zero locus of BGG solutions.

Abstract

We study the standard tractor bundle and the standard cotractor bundle of an almost Grassmann structure: We provide explicit formulae for their splitting operators, first BGG operators as well as prolongation connections. We characterize parallel tractors and cotractors as well as the solutions of the BGG operators in standard geometric terms. Moreover, we describe the geometry canonically endowed on the zero locus of a solution of the first BGG operators.