View source for Pseudo-group structure

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''on a manifold  $  M $''

A maximal atlas  $  A $
of smooth local diffeomorphisms (cf. [[Diffeomorphism]]) from  $  M $
onto a fixed manifold  $  V $, 
all transition functions between them belonging to a given pseudo-group  $  \Gamma $
of local transformations of  $  V $. 
The pseudo-group  $  \Gamma $
is called the defining pseudo-group, and  $  V $
is called the model space. The pseudo-group structure with defining group  $  \Gamma $
is also called a  $  \Gamma $-
structure. More precisely, a set  $  A $
of  $  V $-
valued charts of a manifold  $  M $(
i.e. of diffeomorphisms  $  \phi :  U \rightarrow V $
of open subsets  $  U \subset  M $
onto open subsets  $  \phi ( U) \subset  V $) 
is called a pseudo-group structure if a) any point  $  x \in M $
belongs to the domain of definition of a chart  $  \phi $
of  $  A $; 
b) for any charts  $  \phi :  U \rightarrow V $
and  $  \psi :  W \rightarrow V $
from  $  A $
the transition function  $  \psi \circ \phi  ^ {-1} :  \phi ( U \cap W ) \rightarrow \psi ( U \cap W ) $
is a local transformation from the given pseudo-group  $  \Gamma $; 
and c)  $  A $
is a maximal set of charts satisfying condition b).

===Examples of pseudo-group structures.===

1) A [[Pseudo-group|pseudo-group]]  $  \Gamma $
of transformations of a manifold  $  V $
gives a pseudo-group structure  $  ( V , \Gamma ) $
on  $  V $
whose charts are the local transformations of  $  \Gamma $. 
It is called the standard flat  $  \Gamma $-
structure.

2) Let  $  V = K  ^ {n} $
be an  $  n $-
dimensional vector space over  $  K = \mathbf R , \mathbf C $
or a left module over the skew-field of quaternions  $  K = \mathbf H $, 
and let  $  \Gamma $
be the pseudo-group of local transformations of  $  V $
whose principal linear parts belong to the group  $   \mathop{\rm GL} ( n , K ) $. 
The corresponding  $  \Gamma $-
structure on a manifold  $  M $
is the structure of a smooth manifold if  $  K = \mathbf R $, 
of a complex-analytic manifold if  $  K = \mathbf C $
and of a special quaternionic manifold if  $  K = \mathbf H $.

3) Let  $  \Gamma $
be the pseudo-group of local transformations of a vector space  $  V $
preserving a given tensor  $  S $. 
Specifying a  $  \Gamma $-
structure is equivalent to specifying an integrable (global) tensor field of type  $  S $
on a manifold  $  M $. 
E.g., if  $  S $
is a non-degenerate skew-symmetric  $  2 $-
form, then the  $  \Gamma $-
structure is a [[Symplectic structure|symplectic structure]].

4) Let  $  \Gamma $
be the pseudo-group of local transformations of  $  \mathbf R ^{2n+1} $
that preserve, up to a functional multiplier, the differential  $  1 $-
form

$$ 
d x  ^ {0} +
\sum_{i=1}^ { n } 
x  ^ {2i-1}  d x  ^ {2i} .
$$

Then the  $  \Gamma $-
structure is a [[Contact structure|contact structure]].

5) Let  $  V = G / H $
be a [[Homogeneous space|homogeneous space]] of a Lie group  $  G $, 
and let  $  \Gamma $
be the pseudo-group of local transformations of  $  V $
that can be lifted to transformations of  $  G $. 
Then the  $  \Gamma $-
structure is called the pseudo-group structure determined by the homogeneous space  $  V $. 
Examples of such structures are the structure of a space of constant curvature (in particular, that of a locally Euclidean space), and conformally and projectively flat structures.

Let  $  \Gamma $
be a transitive Lie pseudo-group of transformations of  $  V = \mathbf R  ^ {n} $
of order  $  l $, 
see [[Pseudo-group|Pseudo-group]]. The  $  \Gamma $-
structure  $  A $
on a manifold  $  M $
determines a principal subbundle  $  \pi _ {k} :  B  ^ {k} \rightarrow M $
of the co-frame bundle of arbitrary order  $  k $
on  $  M $, 
consisting of the  $  k $-
jets of charts of  $  A $:

$$ 
B  ^ {k}  = \ 
\{ {j _ {x}  ^ {k} \phi } : {\phi \in A , \phi ( x) = 0 } \}
,\ \ 
\pi _ {k} ( j _ {x}  ^ {k} \phi )  =  x .
$$

The structure group of  $  \pi _ {k} $
is the  $  k $-
th order isotropy group  $  G  ^ {k} ( \Gamma ) $
of  $  \Gamma $, 
which acts on  $  B  ^ {k} $
by the formula

$$ 
j _ {0}  ^ {k} ( a) j _ {x}  ^ {k} \phi  = \ 
j _ {x}  ^ {k} ( a \circ \phi ) .
$$

The bundle  $  \pi _ {k} $
is called the  $  k $-
th structure bundle, or  $  G  ^ {k} ( \Gamma ) $-
structure, determined by the pseudo-group structure  $  A $. 
The bundle  $  \pi _ {l} $, 
with  $  l $
the order of  $  \Gamma $, 
in turn, uniquely determines the pseudo-group structure  $  A $
as the set of charts  $  \phi :  U \rightarrow V $
for which

$$ 
j _ {x}  ^ {l} ( a \circ \phi )  \in  B  ^ {l} \ \ 
\textrm{ if }  a \in \Gamma , a \circ \phi ( x) = 0 .
$$

The geometry of  $  \pi _ {k} $
is characterized by the presence of a canonical  $  G  ^ {k} ( \Gamma ) $-
equivariant  $  1 $-
form  $  \theta  ^ {k} :  T B  ^ {k} \rightarrow V + \mathfrak g  ^ {k} ( V) $
that is horizontal relative to the projection  $  B  ^ {k} \rightarrow B  ^ {k-1} $. 
Here  $  \mathfrak g  ^ {k} ( V) $
is the Lie algebra of the isotropy group  $  G  ^ {k} ( \Gamma ) $. 
The  $  1 $-
form  $  \theta  ^ {k} $
is given by

$$ 
\left .
\theta _ {b  ^ {k}  }  ^ {k} ( \dot{b}   ^ {k} )  = \ 

\frac{d}{dt}

j _ {0}  ^ {k-1} ( \phi _ {t} \circ \phi _ {0}  ^ {-1} )
\right | _ {t = 0 }  ,
$$

where

$$ 
b  ^ {k}  =  j _ {x _ {0}  }  ^ {k} ( \phi _ {0} ) ,\ \ 
\dot{b}   ^ {k}  =  
\frac{d}{dt}
 j _ {x _ {t}  }  ^ {k} ( \phi _ {t} ) ,
$$

$$ 
\phi _ {t}  \in  A ,\  \phi _ {t} ( x _ {t} )  =  0 ,\  t \in [ 0 , \epsilon ] ,
$$

and satisfies a certain Maurer–Cartan structure equation (cf. also [[Maurer–Cartan form|Maurer–Cartan form]]). The Lie algebra of infinitesimal automorphisms of the  $  \Gamma $-
structure can be characterized as the Lie algebra of projectable vector fields on  $  B  ^ {l} $
that preserve the canonical  $  1 $-
form  $  \theta  ^ {l} $.

The basic problem in the theory of pseudo-group structures is the description of pseudo-group structures on manifolds with a defining pseudo-group  $  \Gamma $, 
up to equivalence. Two pseudo-group structures on a manifold are called equivalent if one of them can be reduced to the other by a diffeomorphism of the manifold.

Let  $  \Gamma $
be a globalizing transitive pseudo-group of transformations of a simply-connected manifold  $  V $. 
Any simply-connected manifold with a  $  \Gamma $-
structure  $  A $
admits a mapping  $  \rho :  M \rightarrow V $, 
called a Cartan development, that locally is an isomorphism of  $  \Gamma $-
structures. If  $  A $
has some completeness property, then  $  \rho $
is an isomorphism of  $  \Gamma $-
structures and all  $  \Gamma $-
structures of the type considered are forms of the standard  $  \Gamma $-
structure  $  V $, 
i.e. are obtained from  $  V $
by factorization by a freely-acting discrete automorphism group  $  ( V , \Gamma ) $. 
This is the case, e.g. for (pseudo-)Riemannian structures of constant curvature and for conformally-flat structures on compact manifolds  $  M  ^ {n} $, 
$  n > 2 $.

The theory of deformations, originally developed for complex structures, occupies an important place in the theory of pseudo-group structures. In it one studies problems of the description of non-trivial deformations of a  $  \Gamma $-
structure  $  A $, 
i.e. a family  $  A _ {t} $
of  $  \Gamma $-
structures containing the given  $  \Gamma $-
structure and smoothly depending on a parameter  $  t $, 
modulo trivial deformations. The space of formal infinitesimal non-trivial deformations of a given  $  \Gamma $-
structure is described by the one-dimensional cohomology space  $  H  ^ {1} ( M , \Theta ) $
of  $  M $
with coefficients in the sheaf  $  \Theta $
of germs of infinitesimal automorphisms of  $  A $. 
The  $  \Gamma $-
structure is rigid if this space is trivial. If the two-dimensional cohomology space is trivial,  $  H  ^ {2} ( H , \Theta ) = 0 $, 
one can prove, under certain assumptions, that there exist non-trivial deformations of the  $  \Gamma $-
structure, corresponding to given infinitesimal deformations from  $  H  ^ {1} ( M , \Theta ) $.

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  E. Cartan,   "La géométrie des éspaces Riemanniennes" , ''Mém. Sci. Math.'' , '''9''' , Gauthier-Villars  (1925)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  V. Guillemin,   S. Sternberg,   "Deformation theory of pseudogroup structures" , ''Mem. Amer. Math. Soc.'' , '''64''' , Amer. Math. Soc.  (1966)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  A.S. Pollack,   "The integrability of pseudogroup structures"  ''J. Diff. Geom.'' , '''9''' :  3  (1974)  pp. 355–390</TD></TR><TR><TD valign="top">[4a]</TD> <TD valign="top">  P.A. Griffiths,   "Deformations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075710/p075710145.png" />-structures. Part A: General theory of deformations"  ''Math. Ann.'' , '''155''' :  4  (1964)  pp. 292–315</TD></TR><TR><TD valign="top">[4b]</TD> <TD valign="top">  P.A. Griffiths,   "Deformations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075710/p075710146.png" />-structures. Part B: Deformations of geometric <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p075/p075710/p075710147.png" />-structures"  ''Math. Ann.'' , '''158''' :  5  (1965)  pp. 326–351</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  J.F. Pommaret,   "Théorie des déformations des structures"  ''Ann. Inst. H. Poincaré Nouvelle Sér.'' , '''18'''  (1973)  pp. 285–352  (English abstract)</TD></TR><TR><TD valign="top">[6]</TD> <TD valign="top">  L. Berard Bergery,   J.-P. Bourgignon,   J. Lafontaine,   "Déformations localement triviales des variétés Riemanniennes" , ''Differential geometry'' , ''Proc. Symp. Pure Math.'' , '''27''' , Amer. Math. Soc.  (1975)  pp. 3–32</TD></TR><TR><TD valign="top">[7a]</TD> <TD valign="top">  D.C. Spencer,   "Deformation of structures on manifolds defined by transitive, continuous pseudogroups I. Infinitesimal deformations of structure"  ''Ann. of Math.'' , '''76''' :  2  (1962)  pp. 306–398</TD></TR><TR><TD valign="top">[7b]</TD> <TD valign="top">  D.C. Spencer,   "Deformation of structures on manifolds defined by transitive, continuous pseudogroups II. Deformations of structure"  ''Ann. of Math.'' , '''76''' :  3  (1962)  pp. 399–445</TD></TR></table>

====Comments====
For the topic of classifying spaces for  $  \Gamma $-
structures cf. [[#References|[a2]]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  S. Kobayashi,   K. Nomizu,   "Foundations of differential geometry" , '''1''' , Interscience  (1963)  pp. Chapt. 1</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  A. Haefliger,   "Homotopy and integrability"  J.N. Mordeson (ed.)  et al. (ed.) , ''Structure of arbitrary purely inseparable extension fields'' , ''Lect. notes in math.'' , '''173''' , Springer  (1971)  pp. 133–163</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  J.F. Pommaret,   "Systems of partial differential equations and Lie pseudogroups" , Gordon &amp; Breach  (1978)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  M. Hazewinkel (ed.)  M. Gerstenhaber (ed.) , ''Deformation theory of algebras and structures and applications'' , Kluwer  (1988)</TD></TR></table>