In differential geometry, a quaternionic manifold is a quaternionic analog of a complex manifold. The definition is more complicated and technical than the one for complex manifolds due in part to the noncommutativity of the quaternions and in part to the lack of a suitable calculus of holomorphic functions for quaternions. The most succinct definition uses the language of G-structures on a manifold. Specifically, a quaternionic n-manifold can be defined as a smooth manifold of real dimension 4n equipped with a torsion-free GL ( n , H ) ⋅ H × {\displaystyle \operatorname {GL} (n,\mathbb {H} )\cdot \mathbb {H} ^{\times }} -structure. More naïve, but straightforward, definitions lead to a dearth of examples, and exclude spaces like quaternionic projective space which should clearly be considered as quaternionic manifolds.
Early history Marcel Berger's 1955 paper 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 Bonan and Kraines 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 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.}
Definitions
The enhanced quaternionic general linear group 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
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