quaternionic manifold




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


(Quaternionic manifold)

For n2n \geq 2, a quaternionic manifold is a real 4n-dimensional manifold M with a GL(n,H)H ×\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 MM is called almost quaternionic, and this is just a reduction of the structure group of MM along a Lie group inclusion GL(n,H)GL(4n,R)\text{GL}(n, \mathbf{H}) \hookrightarrow \text{GL}(4n, \mathbf{R}). Note that the multiplicative group of the quaternions H ×\mathbf{H}^{\times} can be normalized, so that the second factor of GL(n,H)H ×\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 GL(n,H)Sp(1)\text{GL} (n,\mathbf{H})\cdot \text{Sp}(1).


Other definitions have been given, based on the following useful but more naïve reasoning: consider two almost complex structures on a smooth 4n4n-manifold MM, say M,I\langle M, I \rangle and M,J\langle M, J \rangle, given the relation {I,J}=0\{ I, J \} =0. Then call the structure M,I,J\langle M, I, J \rangle “almost quaternionic”, and a map φ:M,I,JN,K,F\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 M,IN,K\langle M, I \rangle \rightarrow \langle N, K \rangle and also as a map M,JN,F\langle M, J \rangle \rightarrow \langle N, F \rangle.

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

iu 3=u 4,ju 2=u 4 i \frac{\partial}{\partial u_3} = \frac{\partial }{\partial u _4}, j \frac{\partial }{ \partial u_2} = \frac{\partial }{\partial u_4}
iu 1=u 2,ju 1=u 3i \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,ji, j. Such a complex exists on R 4n\mathbf{R}^{4n} for any nn as an extended Cauchy-Fueter complex consisting of the systems above, repeated for each set of four coordinates. Hence, R 4n\mathbf{R}^{4n} as a smooth manifold can always be endowed with an almost quaternionic structure. On this account of quaternionic structure, a quaternionic nn-manifold is then an almost quaternionic structure on a smooth manifold MM such that MM has an atlas of quaternionic maps, considered with respect to the standard quaternionic structure on R 4n\mathbf{R}^{4n} just described. However, such a definition has serious drawbacks, such as the fact that quaternionic projective nn-space HP n:=Sp(n+1)/Sp(n)Sp(1)\mathbf{H}P^n := \text{Sp}(n + 1)/\text{Sp}(n)\text{Sp}(1) is not a “quaternionic manifold” in this sense for any nn. It is, however, a quaternionic manifold under the official definition above.



(quaternion-Kähler manifolds are quaternionic manifolds)

By definition, a quaternion-Kähler manifold MM has holonomy group contained in the direct product group Sp(n)×\timesSp(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 quat\nabla_\text{quat} of \nabla however is not unique, since quat+𝒮\nabla_\text{quat} + \mathcal{S} is another Sp(n)Sp(1)-preserving connection, where 𝒮\mathcal{S} is a (1, 2)-tensor such that for every pMp \in M, 𝒮(p)\mathcal{S}(p) takes values in the first prolongation of the Lie algebra for the G-structure.




Twistor Space of a Quaternionic Manifold



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:

See also

Last revised on March 30, 2019 at 08:35:15. See the history of this page for a list of all contributions to it.