Characteristic conic connections and torsion-free principal connections

TL;DR

This paper explores the relationship between torsion tensors of principal connections and characteristic conic connections on cone structures, providing conditions for torsion-free connections and classifying certain rational curves.

Contribution

It establishes conditions linking characteristic conic connections to torsion-free principal connections and classifies minimal rational curves with specific VMRT structures.

Findings

Conditions under which characteristic conic connections imply torsion-free principal connections

Complete classification of germs of minimal rational curves with specific VMRTs

Verification of conditions for adjoint varieties of certain Lie algebras

Abstract

We study the relation between torsion tensors of principal connections on G-structures and characteristic conic connections on associated cone structures. We formulate sufficient conditions under which the existence of a characteristic conic connection implies the existence of a torsion-free principal connection. We verify these conditions for adjoint varieties of simple Lie algebras, excluding those of type $\textsf{A}_{\ell \neq 2}$ or $\textsf{C}_{\ell}$. As an application, we give a complete classification of the germs of minimal rational curves whose VMRT at a general point is such an adjoint variety: nontrivial ones come from lines on hyperplane sections of certain Grassmannians or minimal rational curves on wonderful group compactifications.