View source for G-structure

<!--
g0430203.png
$#A+1 = 272 n = 2
$#C+1 = 272 : ~/encyclopedia/old_files/data/G043/G.0403020 \BMI G\EMI\AAhstructure
Automatically converted into TeX, above some diagnostics.
Please remove this comment and the {{TEX|auto}} line below,
if TeX found to be correct.
-->

{{TEX|auto}}
{{TEX|done}}

''on a manifold''

A principal subbundle with structure group  $  G $
of the principal bundle of co-frames on the manifold. More exactly, let  $  \pi _ {k} :  M _ {k} \rightarrow M $
be the principal  $   \mathop{\rm GL}  ^ {k} ( n) $-
bundle of all co-frames of order  $  k $
over an  $  n $-
dimensional manifold  $  M $, 
and let  $  G $
be a subgroup of the general linear group  $   \mathop{\rm GL}  ^ {k} ( n) $
of order  $  k $. 
A submanifold  $  P $
of the manifold of  $  k $-
co-frames  $  M _ {k} $
defines a  $  G $-
structure of order  $  k $, 
$  \pi = \pi _ {k} \mid  _ {P} :  P \rightarrow M $, 
if  $  \pi $
defines a principal  $  G $-
bundle, i.e. the fibres of  $  \pi $
are orbits of  $  G $. 
For example, a section  $  x \mapsto u _ {x}  ^ {k} $
of  $  \pi _ {k} $(
a field of co-frames) defines a  $  G $-
structure  $  P = \{ {gu _ {x}  ^ {k} } : {x \in M,  g \in G } \} $, 
which is called the  $  G $-
structure generated by this field of co-frames. Any  $  G $-
structure is locally generated by a field of co-frames. The  $  G $-
structure over the space  $  V = \mathbf R  ^ {n} $
generated by the field of co-frames  $  x \mapsto j _ {x}  ^ {k} (  \mathop{\rm id} ) $, 
where  $   \mathop{\rm id} :  V \rightarrow V $
is the identity mapping, is called the standard flat  $  G $-
structure.

Let  $  \pi :  P \rightarrow M $
be a  $  G $-
structure. The mapping of the manifold  $  P $
into the point  $  eG \in  \mathop{\rm GL}  ^ {k} ( n)/G $
can be extended to a  $   \mathop{\rm GL}  ^ {k} ( n) $-
equivariant mapping  $  S:  M _ {k} \rightarrow  \mathop{\rm GL}  ^ {k} ( n)/G $, 
which can be considered as a structure of type  $   \mathop{\rm GL}  ^ {k} ( n)/G $
on  $  M $. 
If the homogeneous space  $   \mathop{\rm GL}  ^ {k} ( n)/G $
is imbedded as an orbit in a vector space  $  W $
admitting a linear action of  $   \mathop{\rm GL}  ^ {k} ( n) $, 
then the structure  $  S $
can be considered as a linear structure of type  $  W $; 
this is called the Bernard tensor of the  $  G $-
structure  $  \pi $, 
and is often identified with it. Conversely, let  $  S:  M _ {k} \rightarrow W $
be a linear geometric structure of type  $  W $(
for example, a tensor field), whereby  $  S( M _ {k} ) $
belongs to a single orbit  $   \mathop{\rm GL}  ^ {k} ( n) w _ {0} $
of  $   \mathop{\rm GL}  ^ {k} ( n) $. 
$  P = S  ^ {-} 1 ( w _ {0} ) $
is then a  $  G $-
structure, where  $  G $
is the stabilizer of the point  $  w _ {0} $
in  $   \mathop{\rm GL}  ^ {k} ( n) $, 
and  $  S $
is its Bernard tensor. For example, a Riemannian metric defines an  $  O( n) $-
structure, an almost-symplectic structure defines a  $   \mathop{\rm Sp} ( n/2, \mathbf R ) $-
structure, an almost-complex structure defines a  $   \mathop{\rm GL} ( n/2, \mathbf C ) $-
structure, and a torsion-free connection defines a  $   \mathop{\rm GL} ( n) $-
structure of the second order ( $   \mathop{\rm GL} ( n) $
is considered here as a subgroup of the group  $   \mathop{\rm GL}  ^ {2} ( n) $). 
An affinor (a field of endomorphisms) defines a  $  G $-
structure if and only if it has at all points one and the same Jordan normal form  $  A $, 
where  $  G $
is the centralizer of the matrix  $  A $
in  $   \mathop{\rm GL} ( n) $.

The elements of the manifold  $  M _ {k} $
can be considered as co-frames of order 1 on  $  M _ {k-} 1 $, 
which makes it possible to consider the natural bundle  $  \pi  ^ {k} :  M _ {k} \rightarrow M _ {k-} 1 $
as an  $  N  ^ {k} $-
structure of order one, where  $  N  ^ {k} $
is the kernel of the natural homomorphism  $   \mathop{\rm GL}  ^ {k} ( n) \rightarrow  \mathop{\rm GL}  ^ {k-} 1 ( n) $. 
Every  $  G $-
structure  $  \pi :  P \rightarrow M $
of order  $  k $
has a related sequence of  $  G $-
structures of order one,

$$ 
P  \rightarrow  P _ {-} 1  \rightarrow  P _ {-} 2  \rightarrow \dots \rightarrow  P _ {-} k  =  M,
$$

where  $  P _ {-} i = \pi  ^ {k} ( P _ {-} i+ 1 ) \subset  M _ {k-} i $. 
Consequently, the study of  $  G $-
structures of higher order reduces to the study of  $  G $-
structures of order one. A co-frame  $  u _ {x}  ^ {1} \in M _ {1} $
can be considered as an isomorphism  $  u _ {x}  ^ {1} :  T _ {x} M \rightarrow V $.

The  $  1 $-
form  $  \theta :  TM _ {1} \rightarrow V $, 
assigning to a vector  $  X \in T _ {u _ {x}   ^ {1} } M _ {1} $
the value  $  \theta _ {u _ {x}   ^ {1} } ( X) = u _ {x}  ^ {1} ( \pi _ {1} ) _  \star  X $, 
is called the displacement form. In the local coordinates  $  ( x  ^ {i} , u _ {i}  ^ {a} ) $
of  $  M _ {1} $, 
the form  $  \theta $
is expressed as  $  \theta = u _ {i}  ^ {a}  dx  ^ {i} \otimes e _ {a} $, 
where  $  e _ {a} $
is the standard basis in  $  V $.

The restriction  $  \theta _ {P} $
of  $  \theta $
on a  $  G $-
structure  $  P \subset  M _ {1} $
is called the displacement form of the  $  G $-
structure. It possesses the following properties: 1) strong horizontality:  $  \theta _ {P} ( X) = 0 \iff \pi _  \star  X = 0 $; 
and 2)  $  G $-
equivariance:  $  \theta _ {P} \circ g = g \circ \theta _ {P} $
for any  $  g \in G $.

Using the form  $  \theta _ {P} $
it is possible to characterize the principal bundles with base  $  M $
that are isomorphic to a  $  G $-
structure. Namely, a principal  $  G $-
bundle  $  \pi :  P \rightarrow M $
is isomorphic to a  $  G $-
structure if and only if there are a faithful linear representation  $  \alpha $
of the group  $  G $
in an  $  n $-
dimensional vector space  $  V $, 
$  n =  \mathop{\rm dim}  M $, 
and a  $  V $-
valued strongly-horizontal  $  G $-
equivariant  $  1 $-
form  $  \theta $
on  $  P $. 
Removal of the requirement that the representation  $  \alpha $
be faithful gives the concept of a generalized  $  G $-
structure (of order one) on  $  M $, 
namely a principal  $  G $-
bundle  $  P \rightarrow M $
with a linear representation  $  \alpha :  G \rightarrow  \mathop{\rm GL} ( V) $, 
$   \mathop{\rm dim}  V =  \mathop{\rm dim}  M $, 
and a  $  V $-
valued strongly-horizontal  $  G $-
equivariant  $  1 $-
form  $  \theta $
on  $  P $.

An example of a generalized  $  G $-
structure is the canonical bundle  $  \pi :  P \rightarrow G \setminus  P $
over the homogeneous space  $  G \setminus  P $
of a Lie group  $  P $. 
Here  $  \alpha $
is the [[Isotropy representation|isotropy representation]] of the group  $  G $, 
while  $  \theta $
is defined by the [[Maurer–Cartan form|Maurer–Cartan form]] of  $  P $.

Let  $  \pi :  P \rightarrow M $
be a  $  G $-
structure of order one. The bundle  $  \pi  ^  \prime  :  P ^ { \prime } \rightarrow P $
of  $  1 $-
jets of local sections of  $  \pi $
can be considered as a  $  G ^ { \prime } $-
structure on  $  P $, 
where  $  G ^ { \prime } =  \mathop{\rm Hom} ( V, \mathfrak g ) $
is a commutative group,  $  \mathfrak g $
is the Lie algebra of  $  G $, 
that is linearly represented in the space  $  V \oplus \mathfrak g $
by the formula

$$ 
A( v, X)  = \ 
( v, X+ A( v)),\ \ 
A \in G ^ { \prime } ,\ \ 
v \in V,\ \ 
X \in \mathfrak g ,
$$

and that acts on the manifold  $  P ^ { \prime } $
according to the formula

$$ 
H  \mapsto  AH  =  \{ {l _ {p} A( \theta ( h)) + h } : {A \in G ^ { \prime } , p = \pi ^ { \prime } ( H) , h \in H } \}
,
$$

where  $  l _ {p} $
is the canonical isomorphism of the Lie algebra  $  \mathfrak g $
of the group  $  G $
onto the vertical subspace  $  T _ {p}  ^ {V} P = T _ {p} ( \pi  ^ {-} 1 ( \pi ( p))) $. 
Here the element  $  H \in P ^ { \prime } $
is considered as a horizontal (i.e. complementary to the vertical) subspace in  $  T _ {p} P $. 
It defines a co-frame  $  \theta _ {H}  ^  \prime  :  T _ {p} P  \mathop \rightarrow \limits ^  \approx   \mathfrak g + V $, 
which is defined on a vertical subspace by the mapping  $  l _ {p} $, 
and on a horizontal subspace by the mapping  $  \theta _ {H} = \theta \mid  _ {H} $. 
The vector function  $  C ^ { \prime } :  P ^ { \prime } \rightarrow W =  \mathop{\rm Hom} ( V \wedge V, V) $, 
defined by the formula  $  H \mapsto C _ {H} ^ { \prime } $, 
$  C _ {H} ^ { \prime } ( u, v) = d \theta ( \theta _ {H}  ^ {-} 1 u , \theta _ {H}  ^ {-} 1 v) $, 
is called the torsion function of the  $  G $-
structure  $  \pi $. 
A section  $  s:  x \mapsto H _ {p(} x) $
of the bundle  $  \pi \circ \pi  ^  \prime  :  P ^ { \prime } \rightarrow M $
defines a connection on  $  \pi $, 
while the restriction of the function  $  C ^ { \prime } $
on  $  s( M) $
is a function defining the coordinates of the torsion tensor of this connection relative to the field of co-frames  $  p( x) $.

The mapping  $  C ^ { \prime } :  P ^ { \prime } \rightarrow W $
is  $  G ^ { \prime } $-
equivariant relative to the above-mentioned action of  $  G ^ { \prime } $
on  $  P $
and to the action of  $  G ^ { \prime } $
on  $  W $, 
which is defined by the formula

$$ 
A:  w  \mapsto  Aw  =  w + \delta A,
$$

where  $  \delta :  G ^ { \prime } \rightarrow W $, 
$  ( \delta A)( u, v) = A( u) v - A( v) u $. 
The mapping  $  C:  P \rightarrow G ^ { \prime } \setminus  W $
induced by the mapping  $  C ^ { \prime } $
is called the structure function of the  $  G $-
structure  $  \pi $, 
the vanishing of  $  C $
is equivalent to the existence of a torsion-free connection on  $  \pi $.

The choice of a subspace  $  D \subset  W $
complementary to  $  \delta G ^ { \prime } $
defines a subbundle  $  P  ^ {(} 1) = C ^ { \prime - 1 } ( D) $
of the bundle of co-frames  $  \pi ^ { \prime } :  P ^ { \prime } \rightarrow P $
with structure group  $  G  ^ {(} 1) = G ^ { \prime } \cap  \mathop{\rm Ker}  \delta \cong \mathfrak g \otimes V  ^  \star  \cap V \otimes S  ^ {2} V  ^  \star  \subset  V \otimes V  ^  \star  2 $, 
i.e. a  $  G  ^ {(} 1) $-
structure  $  \pi  ^ {(} 1) = \pi  ^  \prime  \mid  _ {P  ^ {(}  1) } :  P  ^ {(} 1) \rightarrow P $
on  $  P $. 
It is called the first prolongation of the  $  G $-
structure  $  \pi $. 
The  $  i $-
th prolongation  $  \pi  ^ {(} i) :  P  ^ {(} i) \rightarrow P  ^ {(} i- 1) $
is defined by induction as the  $  G  ^ {(} i) $-
structure on  $  P  ^ {(} i- 1) $, 
where the group  $  G  ^ {(} i) $
is isomorphic to the vector group  $  \mathfrak g \otimes S  ^ {i} V  ^  \star  \cap V \otimes S  ^ {i+} 1 V  ^  \star  \subset  V \otimes V  ^ {\star(} i+ 1) $. 
The structure function  $  C  ^ {(} i) $
of the  $  i $-
th prolongation is called the structure function of  $  i $-
th order of the  $  G $-
structure  $  \pi $.

The central problem of the theory of  $  G $-
structures is the local equivalence problem, i.e. the problem of finding necessary and sufficient conditions under which two  $  G $-
structures  $  \pi :  P \rightarrow M $
and  $  \overline \pi \; :  \overline{P}\; \rightarrow \overline{M}\; $
with the same structure group  $  G $
are locally equivalent, i.e. a local diffeomorphism  $  \phi :  M \supset U \rightarrow \overline{U}\; \subset  \overline{M}\; $
of the manifolds  $  M $
and  $  \overline{M}\; $
should exist that induces an isomorphism of  $  G $-
structures over the neighbourhoods  $  U $
and  $  \overline{U}\; $. 
A particular case of this problem is the integrability problem, i.e. the problem of finding necessary and sufficient conditions for the local equivalence of a given  $  G $-
structure and the standard flat  $  G $-
structure. The local equivalence problem can be reformulated as the problem of finding a complete system of local invariants of a  $  G $-
structure.

For an  $  O( n) $-
structure, which is identified with a Riemannian metric, the integrability problem was solved by B. Riemann: Necessary and sufficient conditions for integrability consist in the vanishing of the curvature tensor of the metric. The local equivalence problem was solved by E. Christoffel and R. Lipschitz: A complete system of local invariants of a Riemannian metric consists of its curvature tensor and its successive covariant derivatives (see [[#References|[1]]]).

An approach to solving the equivalence problem is based on the concepts of a prolongation and a structure function. Every  $  G $-
structure  $  \pi :  P \rightarrow M $
of order one with structure group  $  G \subset   \mathop{\rm GL} ( n) $
is connected with a sequence of prolongations

$$ 
\dots \rightarrow  P  ^ {(} i)  \rightarrow  P  ^ {(} i- 1)  \rightarrow \dots \rightarrow  P   \mathop \rightarrow \limits ^  \pi    M,
$$

and a sequence of structure functions  $  C  ^ {(} i) $. 
For an  $  O( n) $-
structure, the structure function  $  C  ^ {(} 0) = C $
on  $  P  ^ {(} 0) = P $
is equal to 0, while the essential parts of the remaining structure functions  $  C  ^ {(} i) $, 
$  i > 0 $, 
are identified with the curvature tensor of the corresponding metric and its successive covariant derivatives. For  $  \pi $
to be integrable it is necessary and sufficient that the structure functions  $  C  ^ {(} 0) \dots C  ^ {(} k) $
be constant, and that their values coincide with the corresponding values of the structure functions of the standard flat  $  G $-
structure (see [[#References|[6]]], [[#References|[8]]], [[#References|[9]]]). The number  $  k $
depends only on the group  $  G $. 
For a broad class of linear groups, especially for all irreducible groups  $  G \subset   \mathop{\rm GL} ( n) $
that do not belong to Berger's list of holonomy groups of spaces with a torsion-free affine connection [[#References|[3]]], one has  $  k= 0 $, 
and for a  $  G $-
structure to be integrable it is necessary and sufficient that the structure function  $  C  ^ {(} 0) $
vanishes, or that a torsion-free linear connection exists, preserving the  $  G $-
structure.

A  $  G $-
structure  $  \pi $
is called a  $  G $-
structure of finite type (equal to  $  k $) 
if  $  G  ^ {(} k- 1) \neq \{ e \} $, 
$  G  ^ {(} k) = \{ e \} $. 
In this case  $  \pi  ^ {(} k) :  P  ^ {(} k) \rightarrow P  ^ {(} k- 1) $
is a field of co-frames (an absolute parallelism), and the automorphism group of the  $  G $-
structure  $  \pi $
is isomorphic to the automorphism group of this parallelism and is a Lie group. The local equivalence problem of these structures reduces to the equivalence problem of absolute parallelisms and has been solved in terms of a finite sequence of structure functions (see [[#References|[2]]]). For a  $  G $-
structure of infinite type, the local equivalence problem remains unsolved in the general case (1984).

Two  $  G $-
structures  $  \pi :  P \rightarrow M $
and  $  \pi  ^  \prime  :  P ^ { \prime } \rightarrow M ^ { \prime } $
are called formally equivalent at the points  $  x \in M $, 
$  x  ^  \prime  \in M ^ { \prime } $
if an isomorphism of the fibres  $  \pi  ^ {-} 1 ( x) \rightarrow \pi  ^ {-} 1 ( x  ^  \prime  ) $
exists that can be continued to an isomorphism of the corresponding fibres of the prolongations  $  P  ^ {(} i) \rightarrow M $
and  $  P ^ { \prime ( i) } \rightarrow M ^ { \prime } $
$  ( i \geq  0) $. 
Examples have been found which demonstrate that if two  $  G $-
structures of class  $  C  ^  \infty  $
are formally equivalent for all pairs  $  ( x, x  ^  \prime  ) \in M \times M ^ { \prime } $, 
then it does not follow, generally speaking, that they are locally equivalent [[#References|[6]]]. In the analytic case, proper subsets  $  S( M) \subset  M $, 
$  S( M ^ { \prime } ) \subset  M ^ { \prime } $
exist, which are countable unions of analytic sets, such that for any  $  x \in M \setminus  S( M) $, 
$  x  ^  \prime  \in M ^ { \prime } \setminus  S( M ^ { \prime } ) $, 
the formal equivalence of two structures  $  P $
and  $  P ^ { \prime } $
at the points  $  x, x  ^  \prime  $
implies that they are locally equivalent [[#References|[7]]].

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  S. Kobayashi,   K. Nomizu,   "Foundations of differential geometry" , '''1''' , Wiley  (1963)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  S. Sternberg,   "Lectures on differential geometry" , Prentice-Hall  (1964)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  M. Berger,   "Sur les groupes d'holonomie homogène des variétés à connexion affine et des variétés riemanniennes"  ''Bull. Soc. Math. France'' , '''83'''  (1955)  pp. 279–330</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  S.S. Chern,   "The geometry of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g043020289.png" />-structures"  ''Bull. Amer. Math. Soc.'' , '''72'''  (1966)  pp. 167–219</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  S. Kobayashi,   "Transformation groups in differential geometry" , Springer  (1972)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  P. Molino,   "Théorie des <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g043020290.png" />-structures: le problème d'Aeequivalence" , Springer  (1977)</TD></TR><TR><TD valign="top">[7]</TD> <TD valign="top">  T. Morimoto,   "Sur le problème d'équivalence des structures géométriques"  ''C.R. Acad. Sci. Paris'' , '''292''' :  1  (1981)  pp. 63–66  (English summary)</TD></TR><TR><TD valign="top">[8]</TD> <TD valign="top">  I.M. Singer,   S. Sternberg,   "The infinite groups of Lie and Cartan. I. The transitive groups"  ''J. d'Anal. Math.'' , '''15'''  (1965)  pp. 1–114</TD></TR><TR><TD valign="top">[9]</TD> <TD valign="top">  A.S. Pollack,   "The integrability problem for pseudogroup structures"  ''J. Diff. Geom.'' , '''9''' :  3  (1974)  pp. 355–390</TD></TR></table>

{{SourceattributionEOM|G-structure}}
[[Category:Mathematics]]