Homogeneous almost quaternion-Hermitian manifolds

TL;DR

This paper classifies compact simply connected homogeneous almost quaternion-Hermitian manifolds with non-zero Euler characteristic, identifying them as Wolf spaces, a product of spheres, or a specific complex quadric.

Contribution

It provides a complete classification of such manifolds, expanding understanding of their geometric structure and properties.

Findings

Identifies all such manifolds as Wolf spaces, $S^2 imes S^2$, or the complex quadric.

Shows that non-vanishing Euler characteristic constrains the manifold's structure.

Contributes to the classification theory of quaternion-Hermitian manifolds.

Abstract

An almost quaternion-Hermitian structure on a Riemannian manifold $(M^{4n},g)$ is a reduction of the structure group of $M$ to $\mathrm{Sp}(n)\mathrm{Sp}(1)\subset \mathrm{SO}(4n)$. In this paper we show that a compact simply connected homogeneous almost quaternion-Hermitian manifold of non-vanishing Euler characteristic is either a Wolf space, or $\mathbb{S}^2\times \mathbb{S}^2$, or the complex quadric $\mathrm{SO}(7)/\mathrm{U}(3)$.