Homogeneous almost 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 $\mathrm{Sp}(n)\mathrm{Sp}(1)\subset \text{ SO }(4n)$. 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 $\text{ SO }(7)/\mathrm{U}(3)$.

Appendix: Root systems

For the basic theory of root systems we refer to [1] and [16].

Definition 4.1

A set $\mathcal R$ of vectors in a Euclidean space $(V,\langle \,\!\cdot ,\,\cdot \!\,\rangle \!)$ is called a root system if it satisfies the following conditions:

• R1: $\mathcal R$ is finite, $\mathrm{span}(\mathcal R )=V$, $0\notin \mathcal R$.

• R2: If $\alpha \in \mathcal R$, then the only multiples of $\alpha$ in $\mathcal R$ are $\pm \alpha$.

• R3: $\frac{2\langle \,\!\alpha ,\,\beta \!\,\rangle }{\langle \,\!\alpha ,\,\alpha \!\,\rangle }\in \mathbb Z$, for all $\alpha , \beta \in \mathcal R$.

• R4: $s_\alpha :\mathcal R \rightarrow \mathcal R$, for all $\alpha \in \mathcal R$ ($s_\alpha$ is the reflection $s_\alpha :V\rightarrow V$, $s_\alpha (v):=v -\frac{2\langle \,\!\alpha ,\,v\!\,\rangle }{\langle \,\!\alpha ,\,\alpha \!\,\rangle }\alpha$).

Let $G$ be a compact semi-simple Lie group with Lie algebra $\mathfrak g$ endowed with an $\mathrm{ad}_\mathfrak{g }$-invariant scalar product. Fix a Cartan sub-algebra $\mathfrak t \subset \mathfrak{g }$ and let $\mathcal R (\mathfrak{g })\subset \mathfrak t ^*$ denote its root system. It is well-known that $\mathcal R (\mathfrak{g })$ satisfies the conditions in Definition 4.1. Conversely, every set of vectors satisfying the conditions in Definition 4.1 is the root system of a unique semi-simple Lie algebra of compact type.

Remark 4.2

(Properties of root systems). Let $\mathcal R$ be a root system. If $\alpha ,\beta \in \mathcal R$ such that $\beta \ne \pm \alpha$ and $\Vert \beta \Vert ^2\ge \Vert \alpha \Vert ^2$, then either $\langle \,\!\alpha ,\,\beta \!\,\rangle =0$ or

$$\begin{aligned} \left( \frac{\Vert \beta \Vert ^2}{\Vert \alpha \Vert ^2}, \frac{2\langle \,\!\alpha ,\,\beta \!\,\rangle }{\langle \,\!\alpha ,\,\alpha \!\,\rangle }\right) \in \{(1,\pm 1),(2,\pm 2),(3,\pm 3)\}. \end{aligned}$$

(10)

In other words, either the scalar product of two roots vanishes, or its absolute value equals half the square length of the longest root. Moreover,

$$\begin{aligned} \beta -\mathrm{sgn}\left( \frac{2\langle \,\alpha ,\,\beta \!\,\rangle }{\langle \,\!\alpha ,\,\alpha \!\,\rangle }\right) k\alpha \in \mathcal R , \quad \text{ for } \text{ all } k\in \mathbb Z , 1\le k\le \biggl |\frac{2\langle \,\!\alpha ,\,\beta \!\,\rangle }{\langle \,\!\alpha ,\,\alpha \!\,\rangle }\biggr |. \end{aligned}$$

(11)

Definition 4.3

([11]) A set $\mathcal P$ of vectors in a Euclidean space $(V,\langle \,\!\cdot ,\,\cdot \!\,\rangle \!)$ is called a root sub-system if it satisfies the conditions R1–R3 from Definition 4.1 and if the set $\overline{\mathcal{P }}$ obtained from $\mathcal P$ by taking all possible reflections is a root system.