Editing G-structure on a manifold (section)

=== Example: Torsion for almost complex structures ===

An example of a ''G''-structure is an [[almost complex structure]], that is, a reduction
of a structure group of an even-dimensional manifold to GL(''n'','''C'''). Such a reduction is uniquely determined by a ''C''<sup>&infin;</sup>-linear endomorphism ''J'' &isin; End(''TM'') such that ''J''<sup>2</sup> = &minus;1. In this situation, the torsion can be computed explicitly as follows.

An easy dimension count shows that

:<math>\Omega^2(TM)= \Omega^{2,0}(TM)\oplus \mathrm{im}(\tau)</math>,

where &Omega;<sup>2,0</sup>(''TM'') is a space of forms ''B'' &isin; &Omega;<sup>2</sup>(''TM'') which satisfy

:<math>B(JX,Y) = B(X, JY) = - J B(X,Y).\,</math>

Therefore, the torsion of an almost complex structure can be considered as an element in 
&Omega;<sup>2,0</sup>(''TM''). It is easy to check that the torsion of an almost complex structure is equal to its [[Nijenhuis tensor]].