# G1 structures on flag manifolds

## TL;DR

This paper classifies invariant G1 almost Hermitian structures on generalized flag manifolds using t-root systems, and also characterizes invariant quasi Kähler structures through these roots.

## Contribution

It introduces the concept of connectedness by triples zero sum and applies it to classify invariant structures on flag manifolds.

## Key findings

- Invariant G1 structures are completely classified using t-root connectedness.
- The paper characterizes invariant quasi Kähler structures in terms of t-roots.
- Proves t-roots are connected by triples zero sum, facilitating classification.

## Abstract

Let $U/K_\Theta$ be a generalized flag manifold, where $K_\Theta$ is the centralizer of a torus in $U$. We study $U$-invariant almost Hermitian structures on $U/K_\Theta$. The classification of these structures are naturally related with the system $R_t$ of t-roots associated to $U/K_\Theta$. We introduced the notion of connectedness by triples zero sum in a general set of linear functional and proved that t-roots are connected by triples zero sum. Using this property, the invariant G1 structures on $U/K_\Theta$ are completely classified. We also study the K\"ahler form and classified the invariant quasi K\"ahler structures on $U/K_\Theta$, in terms of t-roots.