Editing G-structure on a manifold

{{DISPLAYTITLE:''G''-structure on a manifold}}
{{Use American English|date = March 2019}}
{{Short description|Structure group sub-bundle on a tangent frame bundle}}
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 principal ''G''-[[subbundle]] of the [[frame bundle#Tangent frame bundle|tangent frame bundle]] F''M'' (or GL(''M'')) of ''M''.

The notion of ''G''-structures includes various classical structures that can be defined on manifolds, which in some cases are [[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.

Generalising this idea to arbitrary [[principal bundle]]s on topological spaces, one can ask if a principal <math>G</math>-bundle over a [[group (mathematics)|group]] <math>G</math> "comes from" a [[subgroup]] <math>H</math> of <math>G</math>. This is called '''reduction of the structure group''' (to <math>H</math>).

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]].

==Reduction of the structure group==
One can ask if a principal <math>G</math>-bundle over a [[group (mathematics)|group]] <math>G</math> "comes from" a [[subgroup]] <math>H</math> of <math>G</math>. This is called '''reduction of the structure group''' (to <math>H</math>), and makes sense for any map <math>H \to G</math>, which need not be an [[inclusion map]] (despite the terminology).

=== Definition ===
In the following, let <math>X</math> be a [[topological space]], <math>G, H</math> topological groups and a group homomorphism <math>\phi\colon H \to G</math>.

==== In terms of concrete bundles ====
Given a principal <math>G</math>-bundle <math>P</math> over <math>X</math>, a ''reduction of the structure group'' (from <math>G</math> to <math>H</math>) is a ''<math>H</math>''-bundle <math>Q</math> and an isomorphism <math>\phi_Q\colon Q \times_H G \to P</math> of the [[associated bundle]] to the original bundle.

==== In terms of classifying spaces ====
Given a map <math>\pi\colon X \to BG</math>, where <math>BG</math> is the [[classifying space]] for <math>G</math>-bundles, a ''reduction of the structure group'' is a map <math>\pi_Q\colon X \to BH</math> and a homotopy <math>\phi_Q\colon B\phi \circ \pi_Q \to \pi</math>.

=== Properties and examples ===
Reductions of the structure group do not always exist. If they exist, they are usually not essentially unique, since the isomorphism <math>\phi</math> is an important part of the data.

As a concrete example, every even-dimensional real [[vector space]] is isomorphic to the underlying real space of a complex vector space: it admits a [[linear complex structure]]. A real [[vector bundle]] admits an [[almost complex]] structure if and only if it is isomorphic to the underlying real bundle of a complex vector bundle. This is then a reduction along the inclusion ''GL''(''n'','''C''') → ''GL''(2''n'','''R''')

In terms of [[transition map]]s, a ''G''-bundle can be reduced if and only if the transition maps can be taken to have values in ''H''. Note that the term ''reduction'' is misleading: it suggests that ''H'' is a subgroup of ''G'', which is often the case, but need not be (for example for [[spin structure]]s): it's properly called a [[Homotopy lifting property|lifting]].

More abstractly, "''G''-bundles over ''X''" is a [[functor]]<ref>Indeed, it is a [[bifunctor]] in ''G'' and ''X''.</ref> in ''G'': Given a Lie group homomorphism ''H'' → ''G'', one gets a map from ''H''-bundles to ''G''-bundles by [[Induced representation|inducing]] (as above). Reduction of the structure group of a ''G''-bundle ''B'' is choosing an ''H''-bundle whose image is ''B''.

The inducing map from ''H''-bundles to ''G''-bundles is in general neither onto nor one-to-one, so the structure group cannot always be reduced, and when it can, this reduction need not be unique. For example, not every manifold is [[orientable]], and those that are orientable admit exactly two orientations.

If ''H'' is a closed subgroup of ''G'', then there is a natural one-to-one correspondence between reductions of a ''G''-bundle ''B'' to ''H'' and global sections of the [[fiber bundle]] ''B''/''H'' obtained by quotienting ''B'' by the right action of ''H''.  Specifically, the [[fibration]] ''B'' → ''B''/''H'' is a principal ''H''-bundle over ''B''/''H''.  If σ : ''X'' →  ''B''/''H'' is a section, then the [[pullback bundle]] ''B''<sub>H</sub> = σ<sup>−1</sup>''B'' is a reduction of ''B''.<ref>In [[classical field theory]], such a section <math>\sigma</math> describes a classical [[Higgs field (classical)|Higgs field]] ({{cite journal|last1=Sardanashvily|first1=G.|year=2006|title=Geometry of Classical Higgs Fields|journal=International Journal of Geometric Methods in Modern Physics|volume=03|pages=139–148|arxiv=hep-th/0510168|doi=10.1142/S0219887806001065}}).
</ref>

== ''G''-structures ==
Every [[vector bundle]] of dimension <math>n</math> has a canonical <math>GL(n)</math>-bundle, the [[frame bundle]]. In particular, every [[Differentiable manifold|smooth manifold]] has a canonical vector bundle, the [[tangent bundle]]. For a Lie group <math>G</math> and a group homomorphism <math>\phi\colon G \to GL(n)</math>, a <math>G</math>-structure is a reduction of the structure group of the frame bundle to <math>G</math>.

=== Examples ===
The following examples are defined for [[Real vector bundle|real vector bundles]], particularly the [[tangent bundle]] of a [[manifold|smooth manifold]].
{| class="wikitable"
!Group homomorphism
!Group <math>G</math>
!<math>G</math>-structure
!Obstruction
|-
|<math>GL^+(n) < GL(n)</math>
|[[General linear group#real case|General linear group of positive determinant]]
|[[Orientation (manifold)|Orientation]]
|Bundle must be orientable
|-
|<math>SL(n) < GL(n)</math>
|[[Special linear group]]
|[[Volume form]]
|Bundle must be orientable (<math>SL \to GL^+</math> is a [[deformation retract]])
|-
|<math>SL^{\pm}(n) < GL(n)</math>
|Determinant <math>\pm 1</math>
|Pseudo-[[volume form]]
|Always possible
|-
|<math>O(n) < GL(n)</math>
|[[Orthogonal group]]
|[[Riemannian metric]]
|Always possible (<math>O(n)</math> is the [[maximal compact subgroup]], so the inclusion is a deformation retract)
|-
|<math>O(1,n-1) < GL(n)</math>
|[[Indefinite orthogonal group]]
|[[Pseudo Riemannian metric|Pseudo-Riemannian metric]]
|Topological obstruction<ref>It is a [[gravitational field]] in [[gauge gravitation theory]] ({{Cite journal|last1=Sardanashvily|first1=G.|year=2006|title=Gauge gravitation theory from the geometric viewpoint|journal=International Journal of Geometric Methods in Modern Physics|volume=3|issue=1|pages=v–xx|arxiv=gr-qc/0512115|bibcode=2005gr.qc....12115S}})</ref>
|-
|<math>GL(n,\mathbf{C}) < GL(2n,\mathbf{R})</math>
|[[Complex general linear group]]
|[[almost complex manifold|Almost complex structure]]
|Topological obstruction
|-
|<math>GL(n,\mathbf{H})\cdot Sp(1) < GL(4n,\mathbf{R})</math>
|
* <math>GL(n,\mathbf{H})</math>: [[Quaternion|quaternionic]] general linear group acting on <math>\mathbf{H}^n \cong \mathbf{R}^{4n}</math> from the left
* <math>Sp(1)=Spin(3)</math>: group of unit quaternions acting on <math>\mathbf{H}^n</math> from the right
|almost quaternionic structure<ref name=":0">{{harvnb|Besse|1987|loc=§14.61}}</ref>
|Topological obstruction<ref name=":0" />
|-
|<math>GL(k) \times GL(n-k) < GL(n)</math>
|[[General linear group]]
|Decomposition as a [[Whitney sum]] (direct sum) of sub-bundles of rank <math>k</math> and <math>n-k</math>.
|Topological obstruction
|}
Some <math>G</math>-structures are defined in terms of others: Given a Riemannian metric on an oriented manifold, a <math>G</math>-structure for the 2-fold [[covering space|cover]] <math>\mbox{Spin}(n) \to \mbox{SO}(n)</math> is a [[spin manifold|spin structure]]. (Note that the group homomorphism here is ''not'' an inclusion.)

=== Principal bundles ===
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 because the only bundles over contractible spaces are trivial bundles.

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.

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>.

== Integrability conditions and flat ''G''-structures ==
Several structures on manifolds, such as a complex structure, a [[symplectic structure]], or a [[Kähler manifold|Kähler structure]], are ''G''-structures (and thus can be obstructed), but need to satisfy 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 manifold|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 [[2-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]].

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.

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.

== Isomorphism of ''G''-structures ==
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.

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>.

An '''automorphism''' of a ''G''-structure is an isomorphism of a ''G''-structure ''P'' with itself.  Automorphisms arise frequently<ref>{{harvnb|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.

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.

== 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]].

== 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==
* [[G2-structure|G<sub>2</sub>-structure]]

==Notes==
<references/>

==References==
* {{cite book |last1=Besse |first1=Arthur L. |title=Einstein manifolds |series=Ergebnisse der Mathematik und ihrer Grenzgebiete (3) |volume=10 |publisher=[[Springer-Verlag]] |location=Berlin |year=1987 |isbn=3-540-15279-2 |mr=0867684 |others=Reprinted in 2008 |doi=10.1007/978-3-540-74311-8 |author-link1=Arthur Besse |zbl=0613.53001}}
*{{cite journal | author-link=Shiing-Shen Chern | last = Chern | first = Shiing-Shen | year = 1966 | title = The geometry of ''G''-structures | journal = [[Bulletin of the American Mathematical Society]] | volume = 72 | pages = 167&ndash;219 | doi = 10.1090/S0002-9904-1966-11473-8 | issue=2| doi-access = free }}
* {{cite conference | first = Paul | last = Gauduchon | title = Canonical connections for almost-hypercomplex structures | book-title = Complex Analysis and Geometry |volume=366 | series = Pitman Research Notes in Mathematics Series | publisher = Longman | year = 1997 | pages = 123–13 |url=https://books.google.com/books?id=mCyvdD1zLwQC&pg=PA123 |isbn=978-0-582-29276-5}}
* {{cite book | first = Shoshichi | last = Kobayashi | author-link=Shoshichi Kobayashi|title = Transformation Groups in Differential Geometry | series = Classics in Mathematics | publisher = Springer | year = 1972 | isbn = 978-3-540-58659-3 | oclc = 31374337}}
*{{cite book | last = Sternberg | first = Shlomo |author-link=Shlomo Sternberg| year = 1983 | title = Lectures on Differential Geometry | edition = (2nd ed.) | publisher = Chelsea Publishing Co. | location = New York | isbn = 978-0-8218-1385-0 | oclc = 43032711}}
*{{cite journal|title=Reductive G-structures and Lie derivatives|last1=Godina|first1= Marco |last2= Matteucci|first2= Paolo |journal=[[Journal of Geometry and Physics]] |volume=47|issue=1|year=2003|pages=66–86 |doi=10.1016/S0393-0440(02)00174-2|arxiv=math/0201235|bibcode=2003JGP....47...66G|mr=2006228|s2cid=119558088}}

{{Manifolds}}

[[Category:Differential geometry]]
[[Category:Structures on manifolds]]