Examples of minimal $G$-structures induced by the Lee form

TL;DR

This paper investigates conditions under which certain almost Hermitian and almost contact metric structures, characterized by the Lee form, are minimal G-structures within the orthonormal frame bundle of a Riemannian manifold.

Contribution

It provides explicit criteria for minimality of G-structures in specific Gray-Hervella classes using the Lee form, including new results for classes $ ext{W}_4$, $ ext{C}_4$, and $ ext{C}_5$.

Findings

Identifies conditions for minimality in $ ext{W}_4$ and $ ext{C}_5$ classes.

Shows that these classes contain minimal G-structures.

Compares $ ext{C}_4$ class with $ ext{W}_4$ class regarding minimality.

Abstract

We compute the condition of minimality of a G-structure for the Gray-Hervella class $\mathcal{W}_4$ of almost hermitian manifolds and $\mathcal{C}_5$ class of almost contact metric structures. We also consider $\mathcal{C}_4$ class by comparison with the Grey-Hervella class $\mathcal{W}_4$. The common feature is the existence of the Lee form $\theta$ representing these structures. We show that these classes contain minimal G-structures. Here, minimality means minimality of a $G$-structure inside oriented orthonormal frame bundle $SO(M)$ of a Riemannian manifold $M$.