# Harmonic $SU(3)$- and $G_2$-structures via spinors

## TL;DR

This paper provides necessary and sufficient conditions for the harmonicity of $SU(3)$ and $G_2$-structures using spinorial descriptions, with applications to homogeneous spaces.

## Contribution

It introduces a spinorial framework to characterize harmonic $SU(3)$ and $G_2$-structures and applies these results to specific homogeneous spaces.

## Key findings

- Derived explicit harmonicity conditions for $SU(3)$ and $G_2$-structures.
- Connected harmonicity to properties of sections induced by $G$-structures.
- Applied the theoretical results to particular homogeneous spaces.

## Abstract

In this note, using the spinorial description of $SU(3)$ and $G_2$-structures obtained recently by other authors, we give necessary and sufficient conditions for harmonicity of above mentioned structures. We describe obtained results on appropriate homogeneous spaces. Here, harmonicity means harmonicity of the unique section induced by a $G$-structure in consideration.