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User:Fropuff/Drafts/G-structure on a manifold

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Draft in progress for: G-structure on a manifold

Examples

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