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 ![]() |
[4b] | P.A. Griffiths, "Deformations of ![]() ![]() |
[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) |
Pseudo-group structure. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Pseudo-group_structure&oldid=16114