Editing G-structure on a manifold (section)

=== Torsion of a ''G''-structure ===
Associated to any ''G''-structure is a notion of torsion, related to the [[torsion (differential geometry)|torsion]] of a connection.  Note that a given ''G''-structure may admit many different compatible connections which in turn can have different torsions, but in spite of this it is possible to give an independent notion of torsion ''of the G-structure'' as follows.<ref>{{harvnb|Gauduchon|1997}}</ref>

The difference of two adapted connections is a 1-form on ''M'' [[vector-valued differential form|with values in]] the [[adjoint bundle]] Ad<sub>''Q''</sub>. That is to say, the space ''A''<sup>''Q''</sup> of adapted connections is an [[affine space]]
for &Omega;<sup>1</sup>(Ad<sub>''Q''</sub>).

The [[torsion of connection|torsion]] of an adapted connection defines a map

:<math>A^Q \to \Omega^2 (TM)\,</math>

to 2-forms with coefficients in ''TM''. This map is linear; its linearization 

:<math>\tau:\Omega^1(\mathrm{Ad}_Q)\to \Omega^2(TM)\,</math>

is called '''the algebraic torsion map'''. Given two adapted connections &nabla; and &nabla;&prime;, their torsion tensors ''T''<sub>&nabla;</sub>, ''T''<sub>&nabla;&prime;</sub> differ by &tau;(&nabla;&minus;&nabla;&prime;). Therefore, the image of ''T''<sub>&nabla;</sub> in coker(&tau;) is independent from the choice of &nabla;.

The image of ''T''<sub>&nabla;</sub> in coker(&tau;) for any adapted connection &nabla; is called the '''torsion''' of the ''G''-structure. A ''G''-structure is said to be '''torsion-free''' if its torsion vanishes. This happens precisely when ''Q'' admits a torsion-free adapted connection.