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{{Merge|Reduction of the structure group|discuss=Talk:G-structure#Merge proposal|date=August 2012}}
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In [[differential geometry]], a '''''G''-structure''' on an ''n''-[[manifold]] ''M'', for a given [[structure group]]<ref>Which is a [[Lie group]] <math>G \to GL(n,\mathbf{R})</math> mapping to the [[general linear group]] <math>GL(n,\mathbf{R})</math>. This is often but not always a [[Lie subgroup]]; for instance, for a [[spin structure]] the map is a [[covering space]] onto its image.</ref> ''G'', is a ''G''-[[subbundle]] of the [[frame bundle#Tangent frame bundle|tangent frame bundle]] F''M'' (or GL(''M'')) of ''M''.
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The notion of ''G''-structures includes many other structures on manifolds, some of them being defined by [[tensor field]]s. For example, for the [[orthogonal group]], an O(''n'')-structure defines a [[Riemannian metric]], and for the [[special linear group]] an SL(''n'','''R''')-structure is the same as a [[volume form]].  For the [[trivial group]], an {''e''}-structure consists of an [[parallelizable manifold|absolute parallelism]] of the manifold.
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Several structures on manifolds, such as a [[Complex manifold|complex structure]], a [[symplectic structure]], or a [[Kähler manifold|Kähler structure]], are ''G''-structures with an additional [[integrability condition]].
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== Principal bundles and ''G''-structures ==
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Although the theory of [[principal bundle]]s plays an important role in the study of ''G''-structures, the two notions are different. A ''G''-structure is a principal subbundle of the [[frame bundle#Tangent frame bundle|tangent frame bundle]], but the fact that the ''G''-structure bundle ''consists of tangent frames'' is regarded as part of the data. For example, consider two Riemannian metrics on '''R'''<sup>''n''</sup>.  The associated O(''n'')-structures are isomorphic if and only if the metrics are isometric. But, since '''R'''<sup>''n''</sup> is contractible, the underlying O(''n'')-bundles are always going to be isomorphic as principal bundles.
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This fundamental difference between the two theories can be captured by giving an additional piece of data on the underlying ''G''-bundle of a ''G''-structure: the '''[[solder form]]'''.  The solder form is what ties the underlying principal bundle of the ''G''-structure to the local geometry of the manifold itself by specifying a canonical isomorphism of the tangent bundle of ''M'' to an [[associated bundle|associated vector bundle]].  Although the solder form is not a [[connection form]], it can sometimes be regarded as a precursor to one.
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In detail, suppose that ''Q'' is the principal bundle of a ''G''-structure.  If ''Q'' is realized as a reduction of the frame bundle of ''M'', then the solder form is given by the [[pullback (differential geometry)|pullback]] of the [[frame bundle#Solder form|tautological form of the frame bundle]] along the inclusion. Abstractly, if one regards ''Q'' as a principal bundle independently of its realization as a reduction of the frame bundle, then the solder form consists of a representation &rho; of ''G'' on '''R'''<sup>n</sup> and an isomorphism of bundles &theta; : ''TM'' &rarr; ''Q'' &times;<sub>&rho;</sub> '''R'''<sup>n</sup>.
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==Integrability conditions==
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Several structures on manifolds, such as a complex structure, a [[symplectic structure]], or a [[Kähler manifold|Kähler structure]], are ''G''-structures with an additional [[integrability condition]]. Without the corresponding integrability condition, the structure is instead called an "almost" structure, as in an [[almost complex structure]], an [[almost symplectic structure]], or an [[almost Kähler manifold|almost Kähler structure]]


Specifically, a [[symplectic manifold]] structure is a stronger concept than a ''G''-structure for the [[symplectic group]]. A symplectic structure on a manifold is a [[two-form]] ''&omega;'' on ''M'' that is non-degenerate (which is an <math>Sp</math>-structure, or [[almost symplectic structure]]), ''together with'' the extra condition that d''&omega;'' = 0; this latter is called an [[integrability condition]].
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Similarly, [[foliation]]s correspond to ''G''-structures coming from [[block matrix|block matrices]], together with integrability conditions so that the [[Frobenius theorem (differential topology)|Frobenius theorem]] applies.
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== Isomorphism of ''G''-structures ==
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The set of [[diffeomorphism]]s of ''M'' that preserve a ''G''-structure is called the ''[[automorphism group]]'' of that structure. For an O(''n'')-structure they are the group of [[isometry|isometries]] of the Riemannian metric and for an SL(''n'','''R''')-structure volume preserving maps.
 
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Let ''P'' be a ''G''-structure on a manifold ''M'', and ''Q'' a ''G''-structure on a manifold ''N''.  Then an '''isomorphism''' of the ''G''-structures is a diffeomorphism ''f'' : ''M'' &rarr; ''N'' such that the [[pushforward (differential)|pushforward]] of linear frames ''f''<sub>*</sub> : ''FM'' &rarr; ''FN'' restricts to give a mapping of ''P'' into ''Q''.  (Note that it is sufficient that ''Q'' be contained within the image of ''f''<sub>*</sub>.)  The ''G''-structures ''P'' and ''Q'' are '''locally isomorphic''' if ''M'' admits a covering by open sets ''U'' and a family of diffeomorphisms ''f''<sub>U</sub> : ''U'' &rarr; ''f''(''U'') &sub; ''N'' such that ''f''<sub>U</sub> induces an isomorphism of ''P''|<sub>U</sub> &rarr; ''Q''|<sub>''f''(''U'')</sub>.
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An '''automorphism''' of a ''G''-structure is an isomorphism of a ''G''-structure ''P'' with itself. Automorphisms arise frequently<ref>Kobayashi (1972).</ref> in the study of [[transformation group]]s of geometric structures, since many of the important geometric structures on a manifold can be realized as ''G''-structures.
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A '''flat ''G''-structure''' is a ''G''-structure ''P'' having a global section (''V''<sub>1</sub>,...,''V''<sub>n</sub>) consisting of [[Lie derivative|commuting vector fields]].  A ''G''-structure is '''integrable''' (or ''locally flat'') if it is locally isomorphic to a flat ''G''-structure.
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A wide class of [[Cartan's equivalence method|equivalence problems]] can be formulated in the language of ''G''-structures. For example, a pair of Riemannian manifolds are (locally) equivalent if and only if their bundles of [[orthonormal frame]]s are (locally) isomorphic ''G''-structures. In this view, the general procedure for solving an equivalence problem is to construct a system of invariants for the ''G''-structure which are then sufficient to determine whether a pair of ''G''-structures are locally isomorphic or not.
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== Connections on ''G''-structures ==
  <li>[http://globaldialogue2013.go.tz/gdforum/activity http://globaldialogue2013.go.tz/gdforum/activity]</li>
 
 
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'''.
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Concretely speaking, adapted connections can be understood in terms of a [[moving frame]].<ref>Kobayashi (1972) 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
  <li>[http://ukleifeng.co.uk/forum.php?mod=viewthread&tid=57533 http://ukleifeng.co.uk/forum.php?mod=viewthread&tid=57533]</li>
:&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''.
  <li>[http://lab.nqnwebs.com/lavoz_bak/spip.php?article5818/ http://lab.nqnwebs.com/lavoz_bak/spip.php?article5818/]</li>
 
 
=== Torsion of a ''G''-structure ===
</ul>
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>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]].
 
== Higher order ''G''-structures ==
 
Imposing [[integrability condition]]s on a particular ''G''-structure (for instance, with the case of a symplectic form) can be dealt with via the process of [[Cartan's equivalence method|prolongation]]. In such cases, the prolonged ''G''-structure cannot be identified with a ''G''-subbundle of the bundle of linear frames.  In many cases, however, the prolongation is a principal bundle in its own right, and its structure group can be identified with a subgroup of a higher-order [[jet group]]. In which case, it is called a higher order ''G''-structure [Kobayashi]. In general, [[Cartan's equivalence method]] applies to such cases.
 
==See also==
* [[reduction of the structure group]]
 
==Notes==
<references/>
 
==References==
*{{cite journal | authorlink=S. S. Chern | last = Chern | first = S.S. | year = 1966 | title = The geometry of ''G''-structures | journal = Bull. Amer. Math. Soc. | volume = 72 | pages = 167&ndash;219 | doi = 10.1090/S0002-9904-1966-11473-8 | issue=2}}
* {{cite conference | first = P. | last = Gauduchon | title = Canonical connections for almost-hypercomplex structures | booktitle = Complex Analysis and Geometry | series = Pitman Research Notes in Mathematics Series | publisher = Longman | year = 1997 | pages = 123&ndash;136}}
* {{cite book | first = S. | last = Kobayashi | title = Transformation Groups in Differential Geometry | series = Classics in Mathematics | publisher = Springer | year = 1972 | isbn = 3-540-58659-8 | oclc = 31374337}}
*{{cite book | last = Sternberg | first = S. | year = 1983 | title = Lectures on Differential Geometry | edition = (2nd ed.) | publisher = Chelsea Publishing Co. | location = New York | isbn = 0-8218-1385-4 | oclc = 43032711}}
*{{cite journal|title=Reductive G-structures and Lie derivatives|author=Godina M. and Matteucci P.|journal=Journal of Geometry and Physics |volume=47|year=2003|pages=66–86 |doi=10.1016/S0393-0440(02)00174-2}}
 
[[Category:Differential geometry]]
[[Category:Structures on manifolds]]

Latest revision as of 22:01, 3 March 2014

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Can that change again? Sure it could change on some negative development, but because near term technicals go, you would need to move with global liquidity because that's the big game in town. Which i think is the key thing about this market, any time people perceive global markets and liquidity to be in a strong groove, then you can Cheap Nike Air Max Shoes place a positive spin to whatever data you have. I think what's justifying the main trend is global liquidity, post facto.

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