In
differential geometry, a ''G''-structure on an ''n''-
manifold
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a ...
''M'', for a given
structure group
In mathematics, and particularly topology, a fiber bundle (or, in Commonwealth English: fibre bundle) is a space that is a product space, but may have a different topological structure. Specifically, the similarity between a space E an ...
''G'', is a principal ''G''-
subbundle of the
tangent frame bundle F''M'' (or GL(''M'')) of ''M''.
The notion of ''G''-structures includes various classical structures that can be defined on manifolds, which in some cases are
tensor field
In mathematics and physics, a tensor field assigns a tensor to each point of a mathematical space (typically a Euclidean space or manifold). Tensor fields are used in differential geometry, algebraic geometry, general relativity, in the analysis ...
s. For example, for the
orthogonal group, an O(''n'')-structure defines a
Riemannian metric, and for the
special linear group an SL(''n'',R)-structure is the same as a
volume form. For the
trivial group
In mathematics, a trivial group or zero group is a group consisting of a single element. All such groups are isomorphic, so one often speaks of the trivial group. The single element of the trivial group is the identity element and so it is usuall ...
, an -structure consists of an
absolute parallelism of the manifold.
Generalising this idea to arbitrary
principal bundles on topological spaces, one can ask if a principal
-bundle over a
group
A group is a number of persons or things that are located, gathered, or classed together.
Groups of people
* Cultural group, a group whose members share the same cultural identity
* Ethnic group, a group whose members share the same ethnic ide ...
"comes from" a
subgroup
In group theory, a branch of mathematics, given a group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely, ''H'' is a subgrou ...
of
. This is called reduction of the structure group (to
).
Several structures on manifolds, such as a
complex structure, a
symplectic structure
Symplectic geometry is a branch of differential geometry and differential topology that studies symplectic manifolds; that is, differentiable manifolds equipped with a closed, nondegenerate 2-form. Symplectic geometry has its origins in the Ham ...
, or a
Kähler structure, are ''G''-structures with an additional
integrability condition.
Reduction of the structure group
One can ask if a principal
-bundle over a
group
A group is a number of persons or things that are located, gathered, or classed together.
Groups of people
* Cultural group, a group whose members share the same cultural identity
* Ethnic group, a group whose members share the same ethnic ide ...
"comes from" a
subgroup
In group theory, a branch of mathematics, given a group ''G'' under a binary operation ∗, a subset ''H'' of ''G'' is called a subgroup of ''G'' if ''H'' also forms a group under the operation ∗. More precisely, ''H'' is a subgrou ...
of
. This is called reduction of the structure group (to
), and makes sense for any map
, which need not be an
inclusion map (despite the terminology).
Definition
In the following, let
be a
topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called po ...
,
topological groups and a group homomorphism
.
In terms of concrete bundles
Given a principal
-bundle
over
, a ''reduction of the structure group'' (from
to
) is a ''
''-bundle
and an isomorphism
of the
associated bundle In mathematics, the theory of fiber bundles with a structure group G (a topological group) allows an operation of creating an associated bundle, in which the typical fiber of a bundle changes from F_1 to F_2, which are both topological spaces with ...
to the original bundle.
In terms of classifying spaces
Given a map
, where
is the
classifying space
In mathematics, specifically in homotopy theory, a classifying space ''BG'' of a topological group ''G'' is the quotient of a weakly contractible space ''EG'' (i.e. a topological space all of whose homotopy groups are trivial) by a proper free ac ...
for
-bundles, a ''reduction of the structure group'' is a map
and a homotopy
.
Properties and examples
Reductions of the structure group do not always exist. If they exist, they are usually not essentially unique, since the isomorphism
is an important part of the data.
As a concrete example, every even-dimensional real
vector space is isomorphic to the underlying real space of a complex vector space: it admits a
linear complex structure. A real
vector bundle admits an
almost complex structure if and only if it is isomorphic to the underlying real bundle of a complex vector bundle. This is then a reduction along the inclusion ''GL''(''n'',C) → ''GL''(2''n'',R)
In terms of
transition map
In mathematics, particularly topology, one describes a manifold using an atlas. An atlas consists of individual ''charts'' that, roughly speaking, describe individual regions of the manifold. If the manifold is the surface of the Earth, then an ...
s, a ''G''-bundle can be reduced if and only if the transition maps can be taken to have values in ''H''. Note that the term ''reduction'' is misleading: it suggests that ''H'' is a subgroup of ''G'', which is often the case, but need not be (for example for
spin structures): it's properly called a
lifting.
More abstractly, "''G''-bundles over ''X''" is a
functor in ''G'': Given a Lie group homomorphism ''H'' → ''G'', one gets a map from ''H''-bundles to ''G''-bundles by
inducing (as above). Reduction of the structure group of a ''G''-bundle ''B'' is choosing an ''H''-bundle whose image is ''B''.
The inducing map from ''H''-bundles to ''G''-bundles is in general neither onto nor one-to-one, so the structure group cannot always be reduced, and when it can, this reduction need not be unique. For example, not every manifold is
orientable
In mathematics, orientability is a property of some topological spaces such as real vector spaces, Euclidean spaces, surfaces, and more generally manifolds that allows a consistent definition of "clockwise" and "counterclockwise". A space is ...
, and those that are orientable admit exactly two orientations.
If ''H'' is a closed subgroup of ''G'', then there is a natural one-to-one correspondence between reductions of a ''G''-bundle ''B'' to ''H'' and global sections of the
fiber bundle ''B''/''H'' obtained by quotienting ''B'' by the right action of ''H''. Specifically, the
fibration ''B'' → ''B''/''H'' is a principal ''H''-bundle over ''B''/''H''. If σ : ''X'' → ''B''/''H'' is a section, then the
pullback bundle ''B''
H = σ
−1''B'' is a reduction of ''B''.
''G''-structures
Every
vector bundle of dimension
has a canonical
-bundle, the
frame bundle. In particular, every
smooth manifold has a canonical vector bundle, the
tangent bundle. For a Lie group
and a group homomorphism
, a
-structure is a reduction of the structure group of the frame bundle to
.
Examples
The following examples are defined for
real vector bundles, particularly the
tangent bundle of a
smooth manifold.
Some
-structures are defined terms of others: Given a Riemannian metric on an oriented manifold, a
-structure for the 2-fold
cover
Cover or covers may refer to:
Packaging
* Another name for a lid
* Cover (philately), generic term for envelope or package
* Album cover, the front of the packaging
* Book cover or magazine cover
** Book design
** Back cover copy, part of copy ...
is a
spin structure. (Note that the group homomorphism here is ''not'' an inclusion.)
Principal bundles
Although the theory of
principal bundles plays an important role in the study of ''G''-structures, the two notions are different. A ''G''-structure is a principal subbundle of the
tangent frame bundle, but the fact that the ''G''-structure bundle ''consists of tangent frames'' is regarded as part of the data. For example, consider two Riemannian metrics on R
''n''. The associated O(''n'')-structures are isomorphic if and only if the metrics are isometric. But, since R
''n'' is contractible, the underlying O(''n'')-bundles are always going to be isomorphic as principal bundles because the only bundles over contractible spaces are trivial bundles.
This fundamental difference between the two theories can be captured by giving an additional piece of data on the underlying ''G''-bundle of a ''G''-structure: the
solder form
In mathematics, more precisely in differential geometry, a soldering (or sometimes solder form) of a fiber bundle to a smooth manifold is a manner of attaching the fibers to the manifold in such a way that they can be regarded as tangent. Intuitiv ...
. The solder form is what ties the underlying principal bundle of the ''G''-structure to the local geometry of the manifold itself by specifying a canonical isomorphism of the tangent bundle of ''M'' to an
associated vector bundle In mathematics, the theory of fiber bundles with a structure group G (a topological group) allows an operation of creating an associated bundle, in which the typical fiber of a bundle changes from F_1 to F_2, which are both topological spaces with ...
. Although the solder form is not a
connection form In mathematics, and specifically differential geometry, a connection form is a manner of organizing the data of a connection using the language of moving frames and differential forms.
Historically, connection forms were introduced by Élie Carta ...
, it can sometimes be regarded as a precursor to one.
In detail, suppose that ''Q'' is the principal bundle of a ''G''-structure. If ''Q'' is realized as a reduction of the frame bundle of ''M'', then the solder form is given by the
pullback
In mathematics, a pullback is either of two different, but related processes: precomposition and fiber-product. Its dual is a pushforward.
Precomposition
Precomposition with a function probably provides the most elementary notion of pullback: ...
of the
tautological form of the frame bundle along the inclusion. Abstractly, if one regards ''Q'' as a principal bundle independently of its realization as a reduction of the frame bundle, then the solder form consists of a representation ρ of ''G'' on R
n and an isomorphism of bundles θ : ''TM'' → ''Q'' ×
ρ R
n.
Integrability conditions and flat ''G''-structures
Several structures on manifolds, such as a complex structure, a
symplectic structure
Symplectic geometry is a branch of differential geometry and differential topology that studies symplectic manifolds; that is, differentiable manifolds equipped with a closed, nondegenerate 2-form. Symplectic geometry has its origins in the Ham ...
, or a
Kähler structure, are ''G''-structures (and thus can be obstructed), but need to satisfy an additional
integrability condition. Without the corresponding integrability condition, the structure is instead called an "almost" structure, as in an
almost complex structure, an
almost symplectic structure, or an
almost Kähler structure.
Specifically, a
symplectic manifold structure is a stronger concept than a ''G''-structure for the
symplectic group. A symplectic structure on a manifold is a
2-form
In mathematics, differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many applications, ...
''ω'' on ''M'' that is non-degenerate (which is an
-structure, or almost symplectic structure), ''together with'' the extra condition that d''ω'' = 0; this latter is called an
integrability condition.
Similarly,
foliation
In mathematics (differential geometry), a foliation is an equivalence relation on an ''n''-manifold, the equivalence classes being connected, injectively immersed submanifolds, all of the same dimension ''p'', modeled on the decomposition of ...
s correspond to ''G''-structures coming from
block matrices
In mathematics, a block matrix or a partitioned matrix is a matrix that is '' interpreted'' as having been broken into sections called blocks or submatrices. Intuitively, a matrix interpreted as a block matrix can be visualized as the original ma ...
, together with integrability conditions so that the
Frobenius theorem applies.
A flat ''G''-structure is a ''G''-structure ''P'' having a global section (''V''
1,...,''V''
n) consisting of
commuting vector fields. A ''G''-structure is integrable (or ''locally flat'') if it is locally isomorphic to a flat ''G''-structure.
Isomorphism of ''G''-structures
The set of
diffeomorphisms of ''M'' that preserve a ''G''-structure is called the ''
automorphism group
In mathematics, the automorphism group of an object ''X'' is the group consisting of automorphisms of ''X'' under composition of morphisms. For example, if ''X'' is a finite-dimensional vector space, then the automorphism group of ''X'' is the g ...
'' of that structure. For an O(''n'')-structure they are the group of
isometries of the Riemannian metric and for an SL(''n'',R)-structure volume preserving maps.
Let ''P'' be a ''G''-structure on a manifold ''M'', and ''Q'' a ''G''-structure on a manifold ''N''. Then an isomorphism of the ''G''-structures is a diffeomorphism ''f'' : ''M'' → ''N'' such that the
pushforward of linear frames ''f''
* : ''FM'' → ''FN'' restricts to give a mapping of ''P'' into ''Q''. (Note that it is sufficient that ''Q'' be contained within the image of ''f''
*.) The ''G''-structures ''P'' and ''Q'' are locally isomorphic if ''M'' admits a covering by open sets ''U'' and a family of diffeomorphisms ''f''
U : ''U'' → ''f''(''U'') ⊂ ''N'' such that ''f''
U induces an isomorphism of ''P'',
U → ''Q'',
''f''(''U'').
An automorphism of a ''G''-structure is an isomorphism of a ''G''-structure ''P'' with itself. Automorphisms arise frequently in the study of
transformation groups of geometric structures, since many of the important geometric structures on a manifold can be realized as ''G''-structures.
A wide class of
equivalence problems can be formulated in the language of ''G''-structures. For example, a pair of Riemannian manifolds are (locally) equivalent if and only if their bundles of
orthonormal frame
In Riemannian geometry and relativity theory, an orthonormal frame is a tool for studying the structure of a differentiable manifold equipped with a metric. If ''M'' is a manifold equipped with a metric ''g'', then an orthonormal frame at a point ...
s are (locally) isomorphic ''G''-structures. In this view, the general procedure for solving an equivalence problem is to construct a system of invariants for the ''G''-structure which are then sufficient to determine whether a pair of ''G''-structures are locally isomorphic or not.
Connections on ''G''-structures
Let ''Q'' be a ''G''-structure on ''M''. A
principal connection
In mathematics, and especially differential geometry and gauge theory, a connection is a device that defines a notion of parallel transport on the bundle; that is, a way to "connect" or identify fibers over nearby points. A principal ''G''-connect ...
on the principal bundle ''Q'' induces a connection on any associated vector bundle: in particular on the tangent bundle. A
linear connection ∇ on ''TM'' arising in this way is said to be compatible with ''Q''. Connections compatible with ''Q'' are also called adapted connections.
Concretely speaking, adapted connections can be understood in terms of a
moving frame
In mathematics, a moving frame is a flexible generalization of the notion of an ordered basis of a vector space often used to study the extrinsic differential geometry of smooth manifolds embedded in a homogeneous space.
Introduction
In lay te ...
. Suppose that ''V''
i is a basis of local sections of ''TM'' (i.e., a frame on ''M'') which defines a section of ''Q''. Any connection ∇ determines a system of basis-dependent 1-forms ω via
:∇
X V
i = ω
ij(X)V
j
where, as a matrix of 1-forms, ω ∈ Ω
1(M)⊗gl(''n''). An adapted connection is one for which ω takes its values in the Lie algebra g of ''G''.
Torsion of a ''G''-structure
Associated to any ''G''-structure is a notion of torsion, related to the
torsion
Torsion may refer to:
Science
* Torsion (mechanics), the twisting of an object due to an applied torque
* Torsion of spacetime, the field used in Einstein–Cartan theory and
** Alternatives to general relativity
* Torsion angle, in chemistry
Bi ...
of a connection. Note that a given ''G''-structure may admit many different compatible connections which in turn can have different torsions, but in spite of this it is possible to give an independent notion of torsion ''of the G-structure'' as follows.
The difference of two adapted connections is a 1-form on ''M''
with values in the
adjoint bundle Ad
''Q''. That is to say, the space ''A''
''Q'' of adapted connections is an
affine space
In mathematics, an affine space is a geometric structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping only the properties relate ...
for Ω
1(Ad
''Q'').
The
torsion
Torsion may refer to:
Science
* Torsion (mechanics), the twisting of an object due to an applied torque
* Torsion of spacetime, the field used in Einstein–Cartan theory and
** Alternatives to general relativity
* Torsion angle, in chemistry
Bi ...
of an adapted connection defines a map
:
to 2-forms with coefficients in ''TM''. This map is linear; its linearization
:
is called the algebraic torsion map. Given two adapted connections ∇ and ∇′, their torsion tensors ''T''
∇, ''T''
∇′ differ by τ(∇−∇′). Therefore, the image of ''T''
∇ in coker(τ) is independent from the choice of ∇.
The image of ''T''
∇ in coker(τ) for any adapted connection ∇ is called the torsion of the ''G''-structure. A ''G''-structure is said to be torsion-free if its torsion vanishes. This happens precisely when ''Q'' admits a torsion-free adapted connection.
Example: Torsion for almost complex structures
An example of a ''G''-structure is an
almost complex structure, that is, a reduction
of a structure group of an even-dimensional manifold to GL(''n'',C). Such a reduction is uniquely determined by a ''C''
∞-linear endomorphism ''J'' ∈ End(''TM'') such that ''J''
2 = −1. In this situation, the torsion can be computed explicitly as follows.
An easy dimension count shows that
:
,
where Ω
2,0(''TM'') is a space of forms ''B'' ∈ Ω
2(''TM'') which satisfy
:
Therefore, the torsion of an almost complex structure can be considered as an element in
Ω
2,0(''TM''). It is easy to check that the torsion of an almost complex structure is equal to its
Nijenhuis tensor.
Higher order ''G''-structures
Imposing
integrability conditions on a particular ''G''-structure (for instance, with the case of a symplectic form) can be dealt with via the process of
prolongation
In music theory, prolongation is the process in tonal music through which a pitch, interval, or consonant triad is considered to govern spans of music when not physically sounding. It is a central principle in the music-analytic methodology of ...
. In such cases, the prolonged ''G''-structure cannot be identified with a ''G''-subbundle of the bundle of linear frames. In many cases, however, the prolongation is a principal bundle in its own right, and its structure group can be identified with a subgroup of a higher-order
jet group In mathematics, a jet group is a generalization of the general linear group which applies to Taylor polynomials instead of vectors at a point. A jet group is a group of jets that describes how a Taylor polynomial transforms under changes of co ...
. In which case, it is called a higher order ''G''-structure
obayashi In general,
Cartan's equivalence method In mathematics, Cartan's equivalence method is a technique in differential geometry for determining whether two geometrical structures are the same up to a diffeomorphism. For example, if ''M'' and ''N'' are two Riemannian manifolds with metrics ' ...
applies to such cases.
See also
*
G2-structure
Notes
References
*
*
*
*
*
{{Manifolds
Differential geometry
Structures on manifolds