Editing G-structure on a manifold (section)

== Connections on ''G''-structures ==

Let ''Q'' be a ''G''-structure on ''M''.   A [[connection (principal bundle)|principal connection]] on the principal bundle ''Q'' induces a connection on any associated vector bundle: in particular on the tangent bundle.  A  [[connection (vector bundle)|linear connection]] &nabla; on ''TM'' arising in this way is said to be '''compatible''' with ''Q''. Connections compatible with ''Q'' are also called '''adapted connections'''.

Concretely speaking, adapted connections can be understood in terms of a [[moving frame]].<ref>{{harvnb|Kobayashi|1972|loc=I.4}}</ref>  Suppose that ''V''<sub>i</sub> is a basis of local sections of ''TM'' (i.e., a frame on ''M'') which defines a section of ''Q''.  Any connection &nabla; determines a system of basis-dependent 1-forms &omega; via

:&nabla;<sub>X</sub> V<sub>i</sub> = &omega;<sub>i</sub><sup>j</sup>(X)V<sub>j</sub>

where, as a matrix of 1-forms, &omega; &isin; &Omega;<sup>1</sup>(M)&otimes;'''gl'''(''n'').  An adapted connection is one for which &omega; takes its values in the Lie algebra '''g''' of ''G''.

=== 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.

=== 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]].