Pseudo-group structure

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 $ \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.

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.

5) Let $ V = G / H $ be a 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. 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). 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

[1]	E. Cartan, "La géométrie des éspaces Riemanniennes" , Mém. Sci. Math. , 9 , Gauthier-Villars (1925)
[2]	V. Guillemin, S. Sternberg, "Deformation theory of pseudogroup structures" , Mem. Amer. Math. Soc. , 64 , Amer. Math. Soc. (1966)
[3]	A.S. Pollack, "The integrability of pseudogroup structures" J. Diff. Geom. , 9 : 3 (1974) pp. 355–390
[4a]	P.A. Griffiths, "Deformations of $G$-structures. Part A: General theory of deformations" Math. Ann. , 155 : 4 (1964) pp. 292–315
[4b]	P.A. Griffiths, "Deformations of $G$-structures. Part B: Deformations of geometric $G$-structures" Math. Ann. , 158 : 5 (1965) pp. 326–351
[5]	J.F. Pommaret, "Théorie des déformations des structures" Ann. Inst. H. Poincaré Nouvelle Sér. , 18 (1973) pp. 285–352 (English abstract)
[6]	L. Berard Bergery, J.-P. Bourguignon, J. Lafontaine, "Déformations localement triviales des variétés Riemanniennes" , Differential geometry , Proc. Symp. Pure Math. , 27 , Amer. Math. Soc. (1975) pp. 3–32
[7a]	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
[7b]	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

Comments

For the topic of classifying spaces for $ \Gamma $- structures cf. [a2].

References

[a1]	S. Kobayashi, K. Nomizu, "Foundations of differential geometry" , 1 , Interscience (1963) pp. Chapt. 1
[a2]	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
[a3]	J.F. Pommaret, "Systems of partial differential equations and Lie pseudogroups" , Gordon & Breach (1978)
[a4]	M. Hazewinkel (ed.) M. Gerstenhaber (ed.) , Deformation theory of algebras and structures and applications , Kluwer (1988)