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

Draft in progress for: G-structure on a manifold

Examples

$G$	$\operatorname {dim} (M)$	$G$-structure	Torsion-free	Integrable	Comments
$\operatorname {GL} (n,\mathbb {R} )$	$n$	—			Every n-manifold trivally possesses an integrable $\operatorname {GL} (n,\mathbb {R} )$ structure: the frame bundle itself
$\operatorname {GL} ^{+}(n,\mathbb {R} )$	$n$	An orientation			Possible only if the manifold is orientable.
$\operatorname {SL} (n,\mathbb {R} )$	$n$	A volume form			Possible only if the manifold is orientable.
$\{e\}$	$n$	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.
$\operatorname {O} (n)$	$n$	A Riemannian metric		A flat Riemannian metric	Always possible, since $\operatorname {O} (n)$ is a deformation retract of $\operatorname {GL} (n,\mathbb {R} )$. The existence of the Levi-Civita connection means that every $\operatorname {O} (n)$-structure is torsion-free.
$\operatorname {O} (p,q)$	$p+q$	A pseudo-Riemannian metric		A flat pseudo-Riemannian metric	There is a topological obstruction in this case.
$\operatorname {Sp} (2n,\mathbb {R} )$	$2n$	A non-degenerate 2-form	A symplectic form		The torsion of a $\operatorname {Sp} (2n,\mathbb {R} )$-structure $\omega$ is essentially the exterior derivative $d\omega$, so the structure is torsion-free iff $\omega$ is closed. Darboux's theorem says that every torsion-free $\operatorname {Sp} (2n,\mathbb {R} )$-structure is integrable.
$\operatorname {GL} (n,\mathbb {C} )$	$2n$	An almost complex structure	A complex structure		The torsion of a $\operatorname {GL} (n,\mathbb {C} )$-structure $J$ given by the Nijenhuis tensor $N_{J}$. The Newlander–Nirenberg theorem states that every torsion-free $\operatorname {GL} (n,\mathbb {C} )$-structure is integrable.
$\operatorname {U} (n)$	$2n$	A Hermitian metric	A Kähler metric	A flat Kähler metric	$\operatorname {U} (n)=\operatorname {GL} (n,\mathbb {C} )\cap \operatorname {Sp} (2n,\mathbb {R} )\cap \operatorname {O} (2n)$, so this is a compatible combination of a complex, a symplectic, and an orthogonal structure.
$\operatorname {GL} (n,\mathbb {H} )\cdot \mathbb {H} ^{\times }$	$4n$	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.
$\operatorname {GL} (n,\mathbb {H} )$	$4n$		A hypercomplex structure
$\operatorname {Sp} (n)\cdot \operatorname {Sp} (1)$	$4n$	A quaternion-Hermitian metric	A quaternionic Kähler metric
$\operatorname {Sp} (n)$	$4n$		A hyperkähler metric