Quaternionic manifold

In <a href="/facts/Differential_geometry/c04XmyLu">differential geometry</a>, a quaternionic manifold is a <a href="/facts/Quaternion/T3tnu7mX">quaternionic</a> analog of a <a href="/facts/Complex_manifold/DlvPnpzm">complex manifold</a>. The definition is more complicated and technical than the one for complex manifolds due in part to the <a href="/facts/Noncommutative/WaYxJLxd">noncommutativity</a> of the quaternions and in part to the lack of a suitable calculus of <a href="/facts/Holomorphic_function/kVzthtEf">holomorphic functions</a> for quaternions. The most succinct definition uses the language of <a href="/facts/G-structure_on_a_manifold/NmGVT6v0">G-structures on a manifold</a>. Specifically, a quaternionic n-manifold can be defined as a <a href="/facts/Smooth_manifold/aSnbXKcV">smooth manifold</a> of real dimension 4n equipped with a torsion-free 
 
 
 
 GL
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 n
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 H
 
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 H
 
 
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 {\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 <a href="/facts/Quaternionic_projective_space/HbQDIvj8">quaternionic projective space</a> which should clearly be considered as quaternionic manifolds.

Quaternionic manifold open-in-new

Quaternionic manifold