Marcel Berger's 1955 paper1 on the classification of Riemannian holonomy groups first raised the issue of the existence of non-symmetric manifolds with holonomy Sp(n)·Sp(1).Interesting results were proved in the mid-1960s in pioneering work by Edmond Bonan2 and Kraines3 who have independently proven that any such manifold admits a parallel 4-form Ω {\displaystyle \Omega } .The long-awaited analog of strong Lefschetz theorem was published 4 in 1982 : Ω n − k ∧ ⋀ 2 k T ∗ M = ⋀ 4 n − 2 k T ∗ M . {\displaystyle \Omega ^{n-k}\wedge \bigwedge ^{2k}T^{*}M=\bigwedge ^{4n-2k}T^{*}M.}
If we regard the quaternionic vector space H n ≅ R 4 n {\displaystyle \mathbb {H} ^{n}\cong \mathbb {R} ^{4n}} as a right H {\displaystyle \mathbb {H} } -module, we can identify the algebra of right H {\displaystyle \mathbb {H} } -linear maps with the algebra of n × n {\displaystyle n\times n} quaternionic matrices acting on H n {\displaystyle \mathbb {H} ^{n}} from the left. The invertible right H {\displaystyle \mathbb {H} } -linear maps then form a subgroup GL ( n , H ) {\displaystyle \operatorname {GL} (n,\mathbb {H} )} of GL ( 4 n , R ) {\displaystyle \operatorname {GL} (4n,\mathbb {R} )} . We can enhance this group with the group H × {\displaystyle \mathbb {H} ^{\times }} of nonzero quaternions acting by scalar multiplication on H n {\displaystyle \mathbb {H} ^{n}} from the right. Since this scalar multiplication is R {\displaystyle \mathbb {R} } -linear (but not H {\displaystyle \mathbb {H} } -linear) we have another embedding of H × {\displaystyle \mathbb {H} ^{\times }} into GL ( 4 n , R ) {\displaystyle \operatorname {GL} (4n,\mathbb {R} )} . The group GL ( n , H ) ⋅ H × {\displaystyle \operatorname {GL} (n,\mathbb {H} )\cdot \mathbb {H} ^{\times }} is then defined as the product of these subgroups in GL ( 4 n , R ) {\displaystyle \operatorname {GL} (4n,\mathbb {R} )} . Since the intersection of the subgroups GL ( n , H ) {\displaystyle \operatorname {GL} (n,\mathbb {H} )} and H × {\displaystyle \mathbb {H} ^{\times }} in GL ( 4 n , R ) {\displaystyle \operatorname {GL} (4n,\mathbb {R} )} is their mutual center R × {\displaystyle \mathbb {R} ^{\times }} (the group of scalar matrices with nonzero real coefficients), we have the isomorphism
An almost quaternionic structure on a smooth manifold M {\displaystyle M} is just a GL ( n , H ) ⋅ H × {\displaystyle \operatorname {GL} (n,\mathbb {H} )\cdot \mathbb {H} ^{\times }} -structure on M {\displaystyle M} . Equivalently, it can be defined as a subbundle H {\displaystyle H} of the endomorphism bundle End ( T M ) {\displaystyle \operatorname {End} (TM)} such that each fiber H x {\displaystyle H_{x}} is isomorphic (as a real algebra) to the quaternion algebra H {\displaystyle \mathbb {H} } . The subbundle H {\displaystyle H} is called the almost quaternionic structure bundle. A manifold equipped with an almost quaternionic structure is called an almost quaternionic manifold.
The quaternion structure bundle H {\displaystyle H} naturally admits a bundle metric coming from the quaternionic algebra structure, and, with this metric, H {\displaystyle H} splits into an orthogonal direct sum of vector bundles H = L ⊕ E {\displaystyle H=L\oplus E} where L {\displaystyle L} is the trivial line bundle through the identity operator, and E {\displaystyle E} is a rank-3 vector bundle corresponding to the purely imaginary quaternions. Neither the bundles H {\displaystyle H} or E {\displaystyle E} are necessarily trivial.
The unit sphere bundle Z = S ( E ) {\displaystyle Z=S(E)} inside E {\displaystyle E} corresponds to the pure unit imaginary quaternions. These are endomorphisms of the tangent spaces that square to −1. The bundle Z {\displaystyle Z} is called the twistor space of the manifold M {\displaystyle M} , and its properties are described in more detail below. Local sections of Z {\displaystyle Z} are (locally defined) almost complex structures. There exists a neighborhood U {\displaystyle U} of every point x {\displaystyle x} in an almost quaternionic manifold M {\displaystyle M} with an entire 2-sphere of almost complex structures defined on U {\displaystyle U} . One can always find I , J , K ∈ Γ ( Z | U ) {\displaystyle I,J,K\in \Gamma (Z|_{U})} such that
Note, however, that none of these operators may be extendable to all of M {\displaystyle M} . That is, the bundle Z {\displaystyle Z} may admit no global sections (e.g. this is the case with quaternionic projective space H P n {\displaystyle \mathbb {HP} ^{n}} ). This is in marked contrast to the situation for complex manifolds, which always have a globally defined almost complex structure.
A quaternionic structure on a smooth manifold M {\displaystyle M} is an almost quaternionic structure Q {\displaystyle Q} which admits a torsion-free affine connection ∇ {\displaystyle \nabla } preserving Q {\displaystyle Q} . Such a connection is never unique, and is not considered to be part of the quaternionic structure. A quaternionic manifold is a smooth manifold M {\displaystyle M} together with a quaternionic structure on M {\displaystyle M} .
A hypercomplex manifold is a quaternionic manifold with a torsion-free GL ( n , H ) {\displaystyle \operatorname {GL} (n,\mathbb {H} )} -structure. The reduction of the structure group to GL ( n , H ) {\displaystyle \operatorname {GL} (n,\mathbb {H} )} is possible if and only if the almost quaternionic structure bundle H ⊂ End ( T M ) {\displaystyle H\subset \operatorname {End} (TM)} is trivial (i.e. isomorphic to M × H {\displaystyle M\times \mathbb {H} } ). An almost hypercomplex structure corresponds to a global frame of H {\displaystyle H} , or, equivalently, triple of almost complex structures I , J {\displaystyle I,J} , and K {\displaystyle K} such that
A hypercomplex structure is an almost hypercomplex structure such that each of I , J {\displaystyle I,J} , and K {\displaystyle K} are integrable.
A quaternionic Kähler manifold is a quaternionic manifold with a torsion-free Sp ( n ) ⋅ Sp ( 1 ) {\displaystyle \operatorname {Sp} (n)\cdot \operatorname {Sp} (1)} -structure.
A hyperkähler manifold is a quaternionic manifold with a torsion-free Sp ( n ) {\displaystyle \operatorname {Sp} (n)} -structure. A hyperkähler manifold is simultaneously a hypercomplex manifold and a quaternionic Kähler manifold.
Given a quaternionic n {\displaystyle n} -manifold M {\displaystyle M} , the unit 2-sphere subbundle Z = S ( E ) {\displaystyle Z=S(E)} corresponding to the pure unit imaginary quaternions (or almost complex structures) is called the twistor space of M {\displaystyle M} . It turns out that, when n ≥ 2 {\displaystyle n\geq 2} , there exists a natural complex structure on Z {\displaystyle Z} such that the fibers of the projection Z → M {\displaystyle Z\to M} are isomorphic to C P 1 {\displaystyle \mathbb {CP} ^{1}} . When n = 1 {\displaystyle n=1} , the space Z {\displaystyle Z} admits a natural almost complex structure, but this structure is integrable only if the manifold is self-dual. It turns out that the quaternionic geometry on M {\displaystyle M} can be reconstructed entirely from holomorphic data on Z {\displaystyle Z} .
The twistor space theory gives a method of translating problems on quaternionic manifolds into problems on complex manifolds, which are much better understood, and amenable to methods from algebraic geometry. Unfortunately, the twistor space of a quaternionic manifold can be quite complicated, even for simple spaces like H n {\displaystyle \mathbb {H} ^{n}} .
Berger, Marcel (1955). "Sur les groups d'holonomie des variétés à connexion affine et des variétés riemanniennes" (PDF). Bull. Soc. Math. France. 83: 279–330. doi:10.24033/bsmf.1464. http://www.numdam.org/article/BSMF_1955__83__279_0.pdf ↩
Bonan, Edmond (1965). "Structure presque quaternale sur une variété differentiable". Comptes Rendus de l'Académie des Sciences. 261: 5445–8. ↩
Kraines, Vivian Yoh (1966). "Topology of quaternionic manifolds" (PDF). Transactions of the American Mathematical Society. 122 (2): 357–367. doi:10.1090/S0002-9947-1966-0192513-X. JSTOR 1994553. https://www.ams.org/tran/1966-122-02/S0002-9947-1966-0192513-X/S0002-9947-1966-0192513-X.pdf ↩
Bonan, Edmond (1982). "Sur l'algèbre extérieure d'une variété presque hermitienne quaternionique". Comptes Rendus de l'Académie des Sciences. 295: 115–118. ↩