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)