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 , a quaternionic manifold is a real 4n-dimensional manifold M with a -structure which admits a torsion-free connection (i.e. is integrable as a G-structure).
When does not exist is called almost quaternionic, and this is just a reduction of the structure group of along a Lie group inclusion . Note that the multiplicative group of the quaternions can be normalized, so that the second factor of is isomorphic to SU(2), or isomorphically again the first quaternionic unitary group Sp(1). Hence one occasionally finds the structure group reduction written .
Other definitions have been given, based on the following useful but more naïve reasoning: consider two almost complex structures on a smooth -manifold , say and , given the relation . Then call the structure “almost quaternionic”, and a map of almost quaternionic structures “quaternionic” if it is separately complex analytic as a map and also as a map .
When is given a quaternionic multiplication with anti-commuting imaginary units and , this is actually equivalent to satisfying the Cauchy-Feuter? complex:
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 . Such a complex exists on for any as an extended Cauchy-Fueter complex consisting of the systems above, repeated for each set of four coordinates. Hence, as a smooth manifold can always be endowed with an almost quaternionic structure. On this account of quaternionic structure, a quaternionic -manifold is then an almost quaternionic structure on a smooth manifold such that has an atlas of quaternionic maps, considered with respect to the standard quaternionic structure on just described. However, such a definition has serious drawbacks, such as the fact that quaternionic projective -space is not a “quaternionic manifold” in this sense for any . It is, however, a quaternionic manifold under the official definition above.
(quaternion-Kähler manifolds are quaternionic manifolds)
By definition, a quaternion-Kähler manifold has holonomy group contained in the direct product group Sp(n)Sp(1), admitting an extension of the Levi-Civita connection on the holonomy bundle as torsion-free. Thus a quaternion-Kähler manifold is automatically quaternionic.
Such extension of however is not unique, since is another Sp(n)Sp(1)-preserving connection, where is a (1, 2)-tensor such that for every , takes values in the first prolongation of the Lie algebra for the G-structure.
(…)
(…)
biquaternionic manifold?, hypercomplex manifold,
Book references include:
Arthur Besse, Einstein Manifolds, Springer-Verlag 1987.
Dominic Joyce, Compact Manifolds with Special Holonomy, Oxford University Press, 2000.
Classical references:
See also
In terms of G-structure:
Dmitri V. Alekseevsky, 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 (1993) Volume: 4, Issue: 1, page 43-52 (dml:244082)
Dmitri V. Alekseevsky, Stefano Marchiafava, Quaternionic-like structures on a manifold: Note II. Automorphism groups and their interrelations, Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni (1993) Volume: 4, Issue: 1, page 53-61 (dml:244299)
Dmitry V. Alekseevsky, Quaternionic structures on a manifold and subordinated structures, Annali di Matematica pura ed applicata 171, 205–273 (1996) (doi:10.1007/BF01759388)
Holonomy, connections, and twistor spaces:
Stefan Ivanov, Ivan Minchev, Simeon Zamkovoy, Twistors of Almost Quaternionic Manifolds
Gueo Grantcharov and Yat Sun Poon, Geometry of Hyper-Kahler Connections with Torsion
Last revised on July 15, 2020 at 18:55:19. See the history of this page for a list of all contributions to it.