Pseudo-group structure

on a manifold

A maximal atlas of smooth local diffeomorphisms (cf. Diffeomorphism) from onto a fixed manifold , all transition functions between them belonging to a given pseudo-group of local transformations of . The pseudo-group is called the defining pseudo-group, and is called the model space. The pseudo-group structure with defining group is also called a -structure. More precisely, a set of -valued charts of a manifold (i.e. of diffeomorphisms of open subsets onto open subsets ) is called a pseudo-group structure if a) any point belongs to the domain of definition of a chart of ; b) for any charts and from the transition function is a local transformation from the given pseudo-group ; and c) is a maximal set of charts satisfying condition b).

Examples of pseudo-group structures.

1) A pseudo-group of transformations of a manifold gives a pseudo-group structure on whose charts are the local transformations of . It is called the standard flat -structure.

2) Let be an -dimensional vector space over or a left module over the skew-field of quaternions , and let be the pseudo-group of local transformations of whose principal linear parts belong to the group . The corresponding -structure on a manifold is the structure of a smooth manifold if , of a complex-analytic manifold if and of a special quaternionic manifold if .

3) Let be the pseudo-group of local transformations of a vector space preserving a given tensor . Specifying a -structure is equivalent to specifying an integrable (global) tensor field of type on a manifold . E.g., if is a non-degenerate skew-symmetric -form, then the -structure is a symplectic structure.

4) Let be the pseudo-group of local transformations of that preserve, up to a functional multiplier, the differential -form

Then the -structure is a contact structure.

5) Let be a homogeneous space of a Lie group , and let be the pseudo-group of local transformations of that can be lifted to transformations of . Then the -structure is called the pseudo-group structure determined by the homogeneous space . 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 be a transitive Lie pseudo-group of transformations of of order , see Pseudo-group. The -structure on a manifold determines a principal subbundle of the co-frame bundle of arbitrary order on , consisting of the -jets of charts of :

The structure group of is the -th order isotropy group of , which acts on by the formula

The bundle is called the -th structure bundle, or -structure, determined by the pseudo-group structure . The bundle , with the order of , in turn, uniquely determines the pseudo-group structure as the set of charts for which

The geometry of is characterized by the presence of a canonical -equivariant -form that is horizontal relative to the projection . Here is the Lie algebra of the isotropy group . The -form is given by

where

and satisfies a certain Maurer–Cartan structure equation (cf. also Maurer–Cartan form). The Lie algebra of infinitesimal automorphisms of the -structure can be characterized as the Lie algebra of projectable vector fields on that preserve the canonical -form .

The basic problem in the theory of pseudo-group structures is the description of pseudo-group structures on manifolds with a defining pseudo-group , 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 be a globalizing transitive pseudo-group of transformations of a simply-connected manifold . Any simply-connected manifold with a -structure admits a mapping , called a Cartan development, that locally is an isomorphism of -structures. If has some completeness property, then is an isomorphism of -structures and all -structures of the type considered are forms of the standard -structure , i.e. are obtained from by factorization by a freely-acting discrete automorphism group . This is the case, e.g. for (pseudo-)Riemannian structures of constant curvature and for conformally-flat structures on compact manifolds , .

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 -structure , i.e. a family of -structures containing the given -structure and smoothly depending on a parameter , modulo trivial deformations. The space of formal infinitesimal non-trivial deformations of a given -structure is described by the one-dimensional cohomology space of with coefficients in the sheaf of germs of infinitesimal automorphisms of . The -structure is rigid if this space is trivial. If the two-dimensional cohomology space is trivial, , one can prove, under certain assumptions, that there exist non-trivial deformations of the -structure, corresponding to given infinitesimal deformations from .

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 -structures. Part A: General theory of deformations" Math. Ann. , 155 : 4 (1964) pp. 292–315
[4b]	P.A. Griffiths, "Deformations of -structures. Part B: Deformations of geometric -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. Bourgignon, 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 -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)