Quaternionic manifold - Imported Knowlpedia Content

Definition

Let $M$ have real dimension $4n$ with $n\geq2$ . A quaternionic manifold is an almost-quaternionic manifold $(M,Q)$ for which there exists a torsion-free connection $\nabla$ on $TM$ that preserves $Q$ :

\nabla_XA\in\Gamma(Q)

for every vector field $X$ and every local section $A\in\Gamma(Q)$ . Such a $\nabla$ is called a quaternionic connection. Preservation of $Q$ does not require a chosen local admissible triple $(I,J,K)$ to be individually parallel; the connection may rotate that triple within $Q$ . The existence of $\nabla$ , not its choice, is part of the definition.

Integrability and intrinsic torsion

The almost-quaternionic reduction has an intrinsic torsion. In dimensions at least eight, its vanishing is equivalent to the existence of a torsion-free connection preserving $Q$ , which explains the synonym “integrable quaternionic manifold.” Unlike the hypercomplex case, a compatible torsion-free connection is generally not unique. The definition and its twistor interpretation originate in Salamon, “Quaternionic Manifolds”.

Relationship to nearby structures

A hypercomplex triple $(I,J,K)$ determines $Q=\operatorname{span}\{I,J,K\}$ , and its Obata connection makes $Q$ quaternionic. This construction forgets the distinguished global frame of $Q$ .

A quaternion-Kähler manifold of dimension at least eight is quaternionic because its Levi-Civita connection is torsion-free and preserves $Q$ . Conversely, a quaternionic manifold has no preferred metric and need not admit a quaternion-Kähler metric. Thus “quaternionic” is an integrability condition on the $GL(n,\mathbb H)Sp(1)$ -structure, whereas “quaternion-Kähler” is a Riemannian holonomy condition.

Examples and non-examples

Quaternionic projective space $\mathbb H P^n$ with its standard $Q$ is quaternionic; its standard metric is in fact quaternion-Kähler. Every hypercomplex manifold is another example after forgetting its global triple.

An arbitrary almost-quaternionic manifold is a near miss: when its intrinsic torsion is nonzero, no torsion-free preserving connection exists. A connection preserving $Q$ but having torsion also fails the defining condition.

Four-dimensional convention

References

Simon Salamon, “Quaternionic Manifolds,” Symposia Mathematica 26, 1982, 139–151. Stable repository record. Relevant: the torsion-free definition, twistor construction, and dimension-four convention.
Arthur L. Besse, Einstein Manifolds, Springer, 1987. Springer DOI record. Relevant: Chapter 14, quaternionic and quaternion-Kähler manifolds.