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In page Quaternionic manifold:

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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} .