G-structure
on a manifold
A principal subbundle with structure group of the principal bundle of co-frames on the manifold. More exactly, let
be the principal
-bundle of all co-frames of order
over an
-dimensional manifold
, and let
be a subgroup of the general linear group
of order
. A submanifold
of the manifold of
-co-frames
defines a
-structure of order
,
, if
defines a principal
-bundle, i.e. the fibres of
are orbits of
. For example, a section
of
(a field of co-frames) defines a
-structure
, which is called the
-structure generated by this field of co-frames. Any
-structure is locally generated by a field of co-frames. The
-structure over the space
generated by the field of co-frames
, where
is the identity mapping, is called the standard flat
-structure.
Let be a
-structure. The mapping of the manifold
into the point
can be extended to a
-equivariant mapping
, which can be considered as a structure of type
on
. If the homogeneous space
is imbedded as an orbit in a vector space
admitting a linear action of
, then the structure
can be considered as a linear structure of type
; this is called the Bernard tensor of the
-structure
, and is often identified with it. Conversely, let
be a linear geometric structure of type
(for example, a tensor field), whereby
belongs to a single orbit
of
.
is then a
-structure, where
is the stabilizer of the point
in
, and
is its Bernard tensor. For example, a Riemannian metric defines an
-structure, an almost-symplectic structure defines a
-structure, an almost-complex structure defines a
-structure, and a torsion-free connection defines a
-structure of the second order (
is considered here as a subgroup of the group
). An affinor (a field of endomorphisms) defines a
-structure if and only if it has at all points one and the same Jordan normal form
, where
is the centralizer of the matrix
in
.
The elements of the manifold can be considered as co-frames of order 1 on
, which makes it possible to consider the natural bundle
as an
-structure of order one, where
is the kernel of the natural homomorphism
. Every
-structure
of order
has a related sequence of
-structures of order one,
![]() |
where . Consequently, the study of
-structures of higher order reduces to the study of
-structures of order one. A co-frame
can be considered as an isomorphism
.
The -form
, assigning to a vector
the value
, is called the displacement form. In the local coordinates
of
, the form
is expressed as
, where
is the standard basis in
.
The restriction of
on a
-structure
is called the displacement form of the
-structure. It possesses the following properties: 1) strong horizontality:
; and 2)
-equivariance:
for any
.
Using the form it is possible to characterize the principal bundles with base
that are isomorphic to a
-structure. Namely, a principal
-bundle
is isomorphic to a
-structure if and only if there are a faithful linear representation
of the group
in an
-dimensional vector space
,
, and a
-valued strongly-horizontal
-equivariant
-form
on
. Removal of the requirement that the representation
be faithful gives the concept of a generalized
-structure (of order one) on
, namely a principal
-bundle
with a linear representation
,
, and a
-valued strongly-horizontal
-equivariant
-form
on
.
An example of a generalized -structure is the canonical bundle
over the homogeneous space
of a Lie group
. Here
is the isotropy representation of the group
, while
is defined by the Maurer–Cartan form of
.
Let be a
-structure of order one. The bundle
of
-jets of local sections of
can be considered as a
-structure on
, where
is a commutative group,
is the Lie algebra of
, that is linearly represented in the space
by the formula
![]() |
and that acts on the manifold according to the formula
![]() |
where is the canonical isomorphism of the Lie algebra
of the group
onto the vertical subspace
. Here the element
is considered as a horizontal (i.e. complementary to the vertical) subspace in
. It defines a co-frame
, which is defined on a vertical subspace by the mapping
, and on a horizontal subspace by the mapping
. The vector function
, defined by the formula
,
, is called the torsion function of the
-structure
. A section
of the bundle
defines a connection on
, while the restriction of the function
on
is a function defining the coordinates of the torsion tensor of this connection relative to the field of co-frames
.
The mapping is
-equivariant relative to the above-mentioned action of
on
and to the action of
on
, which is defined by the formula
![]() |
where ,
. The mapping
induced by the mapping
is called the structure function of the
-structure
, the vanishing of
is equivalent to the existence of a torsion-free connection on
.
The choice of a subspace complementary to
defines a subbundle
of the bundle of co-frames
with structure group
, i.e. a
-structure
on
. It is called the first prolongation of the
-structure
. The
-th prolongation
is defined by induction as the
-structure on
, where the group
is isomorphic to the vector group
. The structure function
of the
-th prolongation is called the structure function of
-th order of the
-structure
.
The central problem of the theory of -structures is the local equivalence problem, i.e. the problem of finding necessary and sufficient conditions under which two
-structures
and
with the same structure group
are locally equivalent, i.e. a local diffeomorphism
of the manifolds
and
should exist that induces an isomorphism of
-structures over the neighbourhoods
and
. A particular case of this problem is the integrability problem, i.e. the problem of finding necessary and sufficient conditions for the local equivalence of a given
-structure and the standard flat
-structure. The local equivalence problem can be reformulated as the problem of finding a complete system of local invariants of a
-structure.
For an -structure, which is identified with a Riemannian metric, the integrability problem was solved by B. Riemann: Necessary and sufficient conditions for integrability consist in the vanishing of the curvature tensor of the metric. The local equivalence problem was solved by E. Christoffel and R. Lipschitz: A complete system of local invariants of a Riemannian metric consists of its curvature tensor and its successive covariant derivatives (see [1]).
An approach to solving the equivalence problem is based on the concepts of a prolongation and a structure function. Every -structure
of order one with structure group
is connected with a sequence of prolongations
![]() |
and a sequence of structure functions . For an
-structure, the structure function
on
is equal to 0, while the essential parts of the remaining structure functions
,
, are identified with the curvature tensor of the corresponding metric and its successive covariant derivatives. For
to be integrable it is necessary and sufficient that the structure functions
be constant, and that their values coincide with the corresponding values of the structure functions of the standard flat
-structure (see [6], [8], [9]). The number
depends only on the group
. For a broad class of linear groups, especially for all irreducible groups
that do not belong to Berger's list of holonomy groups of spaces with a torsion-free affine connection [3], one has
, and for a
-structure to be integrable it is necessary and sufficient that the structure function
vanishes, or that a torsion-free linear connection exists, preserving the
-structure.
A -structure
is called a
-structure of finite type (equal to
) if
,
. In this case
is a field of co-frames (an absolute parallelism), and the automorphism group of the
-structure
is isomorphic to the automorphism group of this parallelism and is a Lie group. The local equivalence problem of these structures reduces to the equivalence problem of absolute parallelisms and has been solved in terms of a finite sequence of structure functions (see [2]). For a
-structure of infinite type, the local equivalence problem remains unsolved in the general case (1984).
Two -structures
and
are called formally equivalent at the points
,
if an isomorphism of the fibres
exists that can be continued to an isomorphism of the corresponding fibres of the prolongations
and
. Examples have been found which demonstrate that if two
-structures of class
are formally equivalent for all pairs
, then it does not follow, generally speaking, that they are locally equivalent [6]. In the analytic case, proper subsets
,
exist, which are countable unions of analytic sets, such that for any
,
, the formal equivalence of two structures
and
at the points
implies that they are locally equivalent [7].
References
[1] | S. Kobayashi, K. Nomizu, "Foundations of differential geometry" , 1 , Wiley (1963) |
[2] | S. Sternberg, "Lectures on differential geometry" , Prentice-Hall (1964) |
[3] | M. Berger, "Sur les groupes d'holonomie homogène des variétés à connexion affine et des variétés riemanniennes" Bull. Soc. Math. France , 83 (1955) pp. 279–330 |
[4] | S.S. Chern, "The geometry of ![]() |
[5] | S. Kobayashi, "Transformation groups in differential geometry" , Springer (1972) |
[6] | P. Molino, "Théorie des ![]() |
[7] | T. Morimoto, "Sur le problème d'équivalence des structures géométriques" C.R. Acad. Sci. Paris , 292 : 1 (1981) pp. 63–66 (English summary) |
[8] | I.M. Singer, S. Sternberg, "The infinite groups of Lie and Cartan. I. The transitive groups" J. d'Anal. Math. , 15 (1965) pp. 1–114 |
[9] | A.S. Pollack, "The integrability problem for pseudogroup structures" J. Diff. Geom. , 9 : 3 (1974) pp. 355–390 |
G-structure. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=G-structure&oldid=16162