# nLab quaternionic manifold

Contents

### Context

#### Riemannian geometry

Riemannian geometry

# Contents

## Definition

There is some variation in the literature on what one calls a “quaternionic manifold‘’. The most general definition, however, encompassing all the others, is:

###### Definition

(Quaternionic manifold)

For $n \geq 2$, a quaternionic manifold is a real 4n-dimensional manifold M with a $\text{GL} (n,\mathbf{H})\cdot \mathbf{H} ^{\times }$-structure which admits a torsion-free connection $\nabla$ (i.e. is integrable as a G-structure).

When $\nabla$ does not exist $M$ is called almost quaternionic, and this is just a reduction of the structure group of $M$ along a Lie group inclusion $\text{GL}(n, \mathbf{H}) \hookrightarrow \text{GL}(4n, \mathbf{R})$. Note that the multiplicative group of the quaternions $\mathbf{H}^{\times}$ can be normalized, so that the second factor of $\text{GL} (n,\mathbf{H})\cdot \mathbf{H} ^{\times }$ is isomorphic to SU(2), or isomorphically again the first quaternionic unitary group Sp(1). Hence one occasionally finds the structure group reduction written $\text{GL} (n,\mathbf{H})\cdot \text{Sp}(1)$.

###### Remark

Other definitions have been given, based on the following useful but more naïve reasoning: consider two almost complex structures on a smooth $4n$-manifold $M$, say $\langle M, I \rangle$ and $\langle M, J \rangle$, given the relation $\{ I, J \} =0$. Then call the structure $\langle M, I, J \rangle$ “almost quaternionic”, and a map $\varphi: \langle M, I, J \rangle \rightarrow \langle N, K, F \rangle$ of almost quaternionic structures “quaternionic” if it is separately complex analytic as a map $\langle M, I \rangle \rightarrow \langle N, K \rangle$ and also as a map $\langle M, J \rangle \rightarrow \langle N, F \rangle$.

When $\mathbf{R}^4$ is given a quaternionic multiplication with anti-commuting imaginary units $i$ and $j$, this is actually equivalent to $\varphi : \mathbf{R}^4 \rightarrow \mathbf{R}^4$ satisfying the Cauchy-Feuter? complex:

$i \frac{\partial}{\partial u_3} = \frac{\partial }{\partial u _4}, j \frac{\partial }{ \partial u_2} = \frac{\partial }{\partial u_4}$
$i \frac{\partial}{\partial u_1} = \frac{\partial }{\partial u_2}, j \frac{\partial }{\partial u_1 } = \frac{\partial }{\partial u_3}$

known as a starting point for defining a notion of “quaternionic holomorphy?”, since the Cauchy-Feuter complex consists of two separate Cauchy-Riemann systems in the imaginary units $i, j$. Such a complex exists on $\mathbf{R}^{4n}$ for any $n$ as an extended Cauchy-Fueter complex consisting of the systems above, repeated for each set of four coordinates. Hence, $\mathbf{R}^{4n}$ as a smooth manifold can always be endowed with an almost quaternionic structure. On this account of quaternionic structure, a quaternionic $n$-manifold is then an almost quaternionic structure on a smooth manifold $M$ such that $M$ has an atlas of quaternionic maps, considered with respect to the standard quaternionic structure on $\mathbf{R}^{4n}$ just described. However, such a definition has serious drawbacks, such as the fact that quaternionic projective $n$-space $\mathbf{H}P^n := \text{Sp}(n + 1)/\text{Sp}(n)\text{Sp}(1)$ is not a “quaternionic manifold” in this sense for any $n$. It is, however, a quaternionic manifold under the official definition above.

## Examples

###### Example

(quaternion-Kähler manifolds are quaternionic manifolds)

By definition, a quaternion-Kähler manifold $M$ has holonomy group contained in the direct product group Sp(n)$\times$Sp(1), admitting an extension of the Levi-Civita connection $\nabla$ on the holonomy bundle as torsion-free. Thus a quaternion-Kähler manifold is automatically quaternionic.

Such extension $\nabla_\text{quat}$ of $\nabla$ however is not unique, since $\nabla_\text{quat} + \mathcal{S}$ is another Sp(n)Sp(1)-preserving connection, where $\mathcal{S}$ is a (1, 2)-tensor such that for every $p \in M$, $\mathcal{S}(p)$ takes values in the first prolongation of the Lie algebra for the G-structure.

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## References

Book references include:

• Arthur Besse, Einstein Manifolds, Springer-Verlag 1987.

• Dominic Joyce, Compact Manifolds with Special Holonomy, Oxford University Press, 2000.

Classical references:

• Edmond Bonan?, “Sur les G-structures de type quaternionien”, Cahiers de topologie et géométrie différentielle catégoriques, tome 9, no 4 (1967), p. 389-463.

• S.M. Salamon, “Differential Geometry of Quaternionic Manifolds”, Annales scientifiques de l’É.N.S. 4e série, tome 19, no 1 (1986), p. 31-55.

• Dmitry Alekseevsky and Stefano Marchiafava?, “Quaternionic-like structures on a manifold - note I. 1-integrability and integrability conditions”, Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni, Serie 9, Vol. 4 (1993), n.1, p. 43–52.

Holonomy, connections, and twistor spaces: