Quaternion-Kähler manifold - Imported Knowlpedia Content

Definition

For $n\geq2$ , a quaternion-Kähler manifold is a Riemannian manifold $(M^{4n},g)$ whose holonomy group is contained in the group $\operatorname{Sp}(n)\operatorname{Sp}(1)$ . Equivalently, $M$ carries a rank-three subbundle $Q\subseteq\operatorname{End}(TM)$ , locally spanned by a quaternionic triple $I,J,K$ , such that $g$ is quaternionic-Hermitian and the Levi-Civita connection preserves $Q$ . It may rotate $I,J,K$ rather than preserve each separately. The definition is Riemannian and does not assert that $M$ is a complex or a Kähler manifold.

Structure and consequences

The preserved bundle $Q$ makes every quaternion-Kähler manifold of dimension at least eight a quaternionic manifold. Its Levi-Civita connection induces a connection on $Q$ , and the associated fundamental four-form is parallel. Such metrics are Einstein; this is a holonomy consequence rather than an additional defining axiom under the higher-dimensional convention Besse, Theorem 14.39.

Examples and nearby structures

Quaternionic projective space $\mathbb H P^n$ with its standard metric is the compact positive-curvature model. Quaternionic hyperbolic space is the corresponding negative-curvature model. A hyperkähler manifold also satisfies the inclusive holonomy condition because $\operatorname{Sp}(n)\subset\operatorname{Sp}(n)\operatorname{Sp}(1)$ , but its Levi-Civita connection preserves a global triple and its Ricci curvature vanishes.

Some authors reserve “quaternion-Kähler” for the nonzero-scalar-curvature case, thereby excluding hyperkähler manifolds; the core uses the inclusive holonomy convention of Salamon, pp. 143–145.

Four-dimensional convention

When $n=1$ , $\operatorname{Sp}(1)\operatorname{Sp}(1)=\operatorname{SO}(4)$ , so the bare holonomy condition imposes no restriction on an oriented Riemannian four-manifold. Authors usually define a four-dimensional quaternion-Kähler manifold instead to be an Einstein self-dual manifold, with “self-dual” versus “anti-self-dual” depending on the orientation convention. This exceptional convention is not included in the core definition.

References

Arthur L. Besse, Einstein Manifolds, Springer, 1987. Springer DOI record. Relevant: Chapter 14, especially Theorem 14.39.
Simon Salamon, “Quaternionic Kähler Manifolds,” Inventiones Mathematicae 67 (1982), 143–171. DOI record. Relevant: pp. 143–145 for the holonomy groups and dimensional convention.