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''on a manifold''
 
''on a manifold''
  
A principal subbundle with structure group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g0430203.png" /> of the principal bundle of co-frames on the manifold. More exactly, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g0430204.png" /> be the principal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g0430205.png" />-bundle of all co-frames of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g0430206.png" /> over an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g0430207.png" />-dimensional manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g0430208.png" />, and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g0430209.png" /> be a subgroup of the general linear group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g04302010.png" /> of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g04302011.png" />. A submanifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g04302012.png" /> of the manifold of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g04302013.png" />-co-frames <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g04302014.png" /> defines a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g04302015.png" />-structure of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g04302016.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g04302017.png" />, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g04302018.png" /> defines a principal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g04302019.png" />-bundle, i.e. the fibres of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g04302020.png" /> are orbits of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g04302021.png" />. For example, a section <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g04302022.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g04302023.png" /> (a field of co-frames) defines a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g04302024.png" />-structure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g04302025.png" />, which is called the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g04302026.png" />-structure generated by this field of co-frames. Any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g04302027.png" />-structure is locally generated by a field of co-frames. The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g04302028.png" />-structure over the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g04302029.png" /> generated by the field of co-frames <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g04302030.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g04302031.png" /> is the identity mapping, is called the standard flat <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g04302033.png" />-structure.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g04302034.png" /> be a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g04302035.png" />-structure. The mapping of the manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g04302036.png" /> into the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g04302037.png" /> can be extended to a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g04302038.png" />-equivariant mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g04302039.png" />, which can be considered as a structure of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g04302040.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g04302041.png" />. If the homogeneous space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g04302042.png" /> is imbedded as an orbit in a vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g04302043.png" /> admitting a linear action of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g04302044.png" />, then the structure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g04302045.png" /> can be considered as a linear structure of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g04302046.png" />; this is called the Bernard tensor of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g04302048.png" />-structure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g04302049.png" />, and is often identified with it. Conversely, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g04302050.png" /> be a linear geometric structure of type <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g04302051.png" /> (for example, a tensor field), whereby <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g04302052.png" /> belongs to a single orbit <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g04302053.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g04302054.png" />. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g04302055.png" /> is then a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g04302056.png" />-structure, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g04302057.png" /> is the stabilizer of the point <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g04302058.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g04302059.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g04302060.png" /> is its Bernard tensor. For example, a Riemannian metric defines an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g04302061.png" />-structure, an almost-symplectic structure defines a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g04302062.png" />-structure, an almost-complex structure defines a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g04302063.png" />-structure, and a torsion-free connection defines a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g04302064.png" />-structure of the second order (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g04302065.png" /> is considered here as a subgroup of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g04302066.png" />). An affinor (a field of endomorphisms) defines a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g04302067.png" />-structure if and only if it has at all points one and the same Jordan normal form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g04302068.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g04302069.png" /> is the centralizer of the matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g04302070.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g04302071.png" />.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g04302072.png" /> can be considered as co-frames of order 1 on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g04302073.png" />, which makes it possible to consider the natural bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g04302074.png" /> as an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g04302075.png" />-structure of order one, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g04302076.png" /> is the kernel of the natural homomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g04302077.png" />. Every <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g04302078.png" />-structure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g04302079.png" /> of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g04302080.png" /> has a related sequence of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g04302081.png" />-structures of order one,
+
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,
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g04302082.png" /></td> </tr></table>
+
$$
 +
P  \rightarrow  P _ {-} 1  \rightarrow  P _ {-} 2  \rightarrow \dots \rightarrow  P _ {-} k  = M,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g04302083.png" />. Consequently, the study of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g04302084.png" />-structures of higher order reduces to the study of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g04302085.png" />-structures of order one. A co-frame <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g04302086.png" /> can be considered as an isomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g04302087.png" />.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g04302088.png" />-form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g04302089.png" />, assigning to a vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g04302090.png" /> the value <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g04302091.png" />, is called the displacement form. In the local coordinates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g04302092.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g04302093.png" />, the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g04302094.png" /> is expressed as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g04302095.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g04302096.png" /> is the standard basis in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g04302097.png" />.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g04302098.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g04302099.png" /> on a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g043020100.png" />-structure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g043020101.png" /> is called the displacement form of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g043020103.png" />-structure. It possesses the following properties: 1) strong horizontality: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g043020104.png" />; and 2) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g043020105.png" />-equivariance: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g043020106.png" /> for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g043020107.png" />.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g043020108.png" /> it is possible to characterize the principal bundles with base <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g043020109.png" /> that are isomorphic to a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g043020110.png" />-structure. Namely, a principal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g043020111.png" />-bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g043020112.png" /> is isomorphic to a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g043020113.png" />-structure if and only if there are a faithful linear representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g043020114.png" /> of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g043020115.png" /> in an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g043020116.png" />-dimensional vector space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g043020117.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g043020118.png" />, and a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g043020119.png" />-valued strongly-horizontal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g043020120.png" />-equivariant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g043020121.png" />-form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g043020122.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g043020123.png" />. Removal of the requirement that the representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g043020124.png" /> be faithful gives the concept of a generalized <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g043020126.png" />-structure (of order one) on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g043020127.png" />, namely a principal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g043020128.png" />-bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g043020129.png" /> with a linear representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g043020130.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g043020131.png" />, and a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g043020132.png" />-valued strongly-horizontal <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g043020133.png" />-equivariant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g043020134.png" />-form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g043020135.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g043020136.png" />.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g043020137.png" />-structure is the canonical bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g043020138.png" /> over the homogeneous space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g043020139.png" /> of a Lie group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g043020140.png" />. Here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g043020141.png" /> is the [[Isotropy representation|isotropy representation]] of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g043020142.png" />, while <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g043020143.png" /> is defined by the [[Maurer–Cartan form|Maurer–Cartan form]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g043020144.png" />.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g043020145.png" /> be a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g043020146.png" />-structure of order one. The bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g043020147.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g043020148.png" />-jets of local sections of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g043020149.png" /> can be considered as a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g043020150.png" />-structure on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g043020151.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g043020152.png" /> is a commutative group, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g043020153.png" /> is the Lie algebra of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g043020154.png" />, that is linearly represented in the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g043020155.png" /> by the formula
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g043020156.png" /></td> </tr></table>
+
$$
 +
A( v, X)  = \
 +
( v, X+ A( v)),\ \
 +
A \in G ^ { \prime } ,\ \
 +
v \in V,\ \
 +
X \in \mathfrak g ,
 +
$$
  
and that acts on the manifold <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g043020157.png" /> according to the formula
+
and that acts on the manifold $  P ^ { \prime } $
 +
according to the formula
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g043020158.png" /></td> </tr></table>
+
$$
 +
H  \mapsto  AH  = \{ {l _ {p} A( \theta ( h)) + h } : {A \in G ^ { \prime } , p = \pi ^ { \prime } ( H) , h \in H } \}
 +
,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g043020159.png" /> is the canonical isomorphism of the Lie algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g043020160.png" /> of the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g043020161.png" /> onto the vertical subspace <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g043020162.png" />. Here the element <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g043020163.png" /> is considered as a horizontal (i.e. complementary to the vertical) subspace in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g043020164.png" />. It defines a co-frame <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g043020165.png" />, which is defined on a vertical subspace by the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g043020166.png" />, and on a horizontal subspace by the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g043020167.png" />. The vector function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g043020168.png" />, defined by the formula <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g043020169.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g043020170.png" />, is called the torsion function of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g043020172.png" />-structure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g043020173.png" />. A section <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g043020174.png" /> of the bundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g043020175.png" /> defines a connection on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g043020176.png" />, while the restriction of the function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g043020177.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g043020178.png" /> is a function defining the coordinates of the torsion tensor of this connection relative to the field of co-frames <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g043020179.png" />.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g043020180.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g043020181.png" />-equivariant relative to the above-mentioned action of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g043020182.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g043020183.png" /> and to the action of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g043020184.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g043020185.png" />, which is defined by the formula
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g043020186.png" /></td> </tr></table>
+
$$
 +
A: w  \mapsto  Aw  = w + \delta A,
 +
$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g043020187.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g043020188.png" />. The mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g043020189.png" /> induced by the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g043020190.png" /> is called the structure function of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g043020192.png" />-structure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g043020193.png" />, the vanishing of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g043020194.png" /> is equivalent to the existence of a torsion-free connection on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g043020195.png" />.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g043020196.png" /> complementary to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g043020197.png" /> defines a subbundle <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g043020198.png" /> of the bundle of co-frames <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g043020199.png" /> with structure group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g043020200.png" />, i.e. a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g043020201.png" />-structure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g043020202.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g043020203.png" />. It is called the first prolongation of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g043020205.png" />-structure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g043020206.png" />. The <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g043020209.png" />-th prolongation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g043020210.png" /> is defined by induction as the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g043020211.png" />-structure on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g043020212.png" />, where the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g043020213.png" /> is isomorphic to the vector group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g043020214.png" />. The structure function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g043020215.png" /> of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g043020216.png" />-th prolongation is called the structure function of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g043020219.png" />-th order of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g043020220.png" />-structure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g043020221.png" />.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g043020222.png" />-structures is the local equivalence problem, i.e. the problem of finding necessary and sufficient conditions under which two <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g043020224.png" />-structures <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g043020225.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g043020226.png" /> with the same structure group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g043020227.png" /> are locally equivalent, i.e. a local diffeomorphism <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g043020228.png" /> of the manifolds <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g043020229.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g043020230.png" /> should exist that induces an isomorphism of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g043020231.png" />-structures over the neighbourhoods <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g043020232.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g043020233.png" />. 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g043020234.png" />-structure and the standard flat <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g043020235.png" />-structure. The local equivalence problem can be reformulated as the problem of finding a complete system of local invariants of a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g043020236.png" />-structure.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g043020237.png" />-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]]]).
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g043020238.png" />-structure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g043020239.png" /> of order one with structure group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g043020240.png" /> is connected with a sequence of prolongations
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g043020241.png" /></td> </tr></table>
+
$$
 +
\dots \rightarrow  P  ^ {(} i)  \rightarrow  P  ^ {(} i- 1)  \rightarrow \dots \rightarrow  P  \mathop \rightarrow \limits ^  \pi    M,
 +
$$
  
and a sequence of structure functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g043020242.png" />. For an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g043020243.png" />-structure, the structure function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g043020244.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g043020245.png" /> is equal to 0, while the essential parts of the remaining structure functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g043020246.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g043020247.png" />, are identified with the curvature tensor of the corresponding metric and its successive covariant derivatives. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g043020248.png" /> to be integrable it is necessary and sufficient that the structure functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g043020249.png" /> be constant, and that their values coincide with the corresponding values of the structure functions of the standard flat <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g043020250.png" />-structure (see [[#References|[6]]], [[#References|[8]]], [[#References|[9]]]). The number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g043020251.png" /> depends only on the group <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g043020252.png" />. For a broad class of linear groups, especially for all irreducible groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g043020253.png" /> that do not belong to Berger's list of holonomy groups of spaces with a torsion-free affine connection [[#References|[3]]], one has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g043020254.png" />, and for a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g043020255.png" />-structure to be integrable it is necessary and sufficient that the structure function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g043020256.png" /> vanishes, or that a torsion-free linear connection exists, preserving the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g043020257.png" />-structure.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g043020258.png" />-structure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g043020259.png" /> is called a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g043020261.png" />-structure of finite type (equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g043020262.png" />) if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g043020263.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g043020264.png" />. In this case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g043020265.png" /> is a field of co-frames (an absolute parallelism), and the automorphism group of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g043020266.png" />-structure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g043020267.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g043020268.png" />-structure of infinite type, the local equivalence problem remains unsolved in the general case (1984).
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g043020269.png" />-structures <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g043020270.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g043020271.png" /> are called formally equivalent at the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g043020273.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g043020274.png" /> if an isomorphism of the fibres <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g043020275.png" /> exists that can be continued to an isomorphism of the corresponding fibres of the prolongations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g043020276.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g043020277.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g043020278.png" />. Examples have been found which demonstrate that if two <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g043020279.png" />-structures of class <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g043020280.png" /> are formally equivalent for all pairs <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g043020281.png" />, then it does not follow, generally speaking, that they are locally equivalent [[#References|[6]]]. In the analytic case, proper subsets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g043020282.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g043020283.png" /> exist, which are countable unions of analytic sets, such that for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g043020284.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g043020285.png" />, the formal equivalence of two structures <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g043020286.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g043020287.png" /> at the points <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/g/g043/g043020/g043020288.png" /> implies that they are locally equivalent [[#References|[7]]].
+
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====
 
====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>
 
<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>

Latest revision as of 19:40, 5 June 2020


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 of the group $ G $, while $ \theta $ is defined by the 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 [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 [6], [8], [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 [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 [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 [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 [7].

References

[1] S. Kobayashi, K. Nomizu, "Foundations of differential geometry" , 1 , Wiley (1963)
[2] S. Sternberg, "Lectures on differential geometry" , Prentice-Hall (1964)
[3] 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
[4] S.S. Chern, "The geometry of -structures" Bull. Amer. Math. Soc. , 72 (1966) pp. 167–219
[5] S. Kobayashi, "Transformation groups in differential geometry" , Springer (1972)
[6] P. Molino, "Théorie des -structures: le problème d'Aeequivalence" , Springer (1977)
[7] T. Morimoto, "Sur le problème d'équivalence des structures géométriques" C.R. Acad. Sci. Paris , 292 : 1 (1981) pp. 63–66 (English summary)
[8] 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
[9] A.S. Pollack, "The integrability problem for pseudogroup structures" J. Diff. Geom. , 9 : 3 (1974) pp. 355–390
How to Cite This Entry:
G-structure. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=G-structure&oldid=47031
This article was adapted from an original article by D.V. Alekseevskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article