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-structure
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Torsion-free
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Integrable
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Comments
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—
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Every n-manifold trivally possesses an integrable structure: the frame bundle itself
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An orientation
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Possible only if the manifold is orientable.
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A volume form
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Possible only if the manifold is orientable.
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A parallelization
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An affine parallelization
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A topological obstruction exists in this case. A parallelization is torsion-free if and only if the given global frame is a holonomic (commuting) frame.
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A Riemannian metric
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A flat Riemannian metric
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Always possible, since is a deformation retract of . The existence of the Levi-Civita connection means that every -structure is torsion-free.
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A pseudo-Riemannian metric
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A flat pseudo-Riemannian metric
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There is a topological obstruction in this case.
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A non-degenerate 2-form
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A symplectic form
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The torsion of a -structure is essentially the exterior derivative , so the structure is torsion-free iff is closed. Darboux's theorem says that every torsion-free -structure is integrable.
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An almost complex structure
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A complex structure
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The torsion of a -structure given by the Nijenhuis tensor . The Newlander–Nirenberg theorem states that every torsion-free -structure is integrable.
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A Hermitian metric
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A Kähler metric
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A flat Kähler metric
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, so this is a compatible combination of a complex, a symplectic, and an orthogonal structure.
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An almost quaternionic structure
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A quaternionic structure
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Unlike the complex case, there is no guarantee of integrability for (torsion-free) quaternionic manifolds. There exist counterexamples.
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A hypercomplex structure
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A quaternion-Hermitian metric
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A quaternionic Kähler metric
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A hyperkähler metric
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