G-structure on a manifold
In
FM (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
Generalising this idea to arbitrary principal bundles on topological spaces, one can ask if a principal -bundle over a group "comes from" a subgroup of . This is called reduction of the structure group (to ).
Several structures on manifolds, such as a
Reduction of the structure group
One can ask if a principal -bundle over a group "comes from" a subgroup 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, 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 to the original bundle.
In terms of classifying spaces
Given a map , where is the classifying space 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
In terms of
More abstractly, "G-bundles over X" is a functor[2] 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
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 BH = σ−1B is a reduction of B.[3]
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
Group homomorphism | Group | -structure | Obstruction |
---|---|---|---|
General linear group of positive determinant | Orientation
|
Bundle must be orientable | |
Special linear group | Volume form | Bundle must be orientable ( is a deformation retract )
| |
Determinant | Pseudo-volume form | Always possible | |
Orthogonal group | Riemannian metric
|
Always possible ( is the maximal compact subgroup, so the inclusion is a deformation retract) | |
Indefinite orthogonal group | Pseudo-Riemannian metric
|
Topological obstruction[4] | |
Complex general linear group
|
Almost complex structure | Topological obstruction | |
|
almost quaternionic structure[5] | Topological obstruction[5] | |
General linear group | Decomposition as a Whitney sum (direct sum) of sub-bundles of rank and .
|
Topological obstruction |
Some -structures are defined in terms of others: Given a Riemannian metric on an oriented manifold, a -structure for the 2-fold cover is a
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 Rn. The associated O(n)-structures are isomorphic if and only if the metrics are isometric. But, since Rn 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. 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. Although the solder form is not a connection form, 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 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 Rn and an isomorphism of bundles θ : TM → Q ×ρ Rn.
Integrability conditions and flat G-structures
Several structures on manifolds, such as a complex structure, a
Specifically, a
Similarly, foliations correspond to G-structures coming from block matrices, together with integrability conditions so that the Frobenius theorem applies.
A flat G-structure is a G-structure P having a global section (V1,...,Vn) 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 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 fU : U → f(U) ⊂ N such that fU 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
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 frames 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 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.[7] Suppose that Vi 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 Vi = ωij(X)Vj
where, as a matrix of 1-forms, ω ∈ Ω1(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
The difference of two adapted connections is a 1-form on M with values in the adjoint bundle AdQ. That is to say, the space AQ of adapted connections is an affine space for Ω1(AdQ).
The
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
An easy dimension count shows that
- ,
where Ω2,0(TM) is a space of forms B ∈ Ω2(TM) which satisfy
Therefore, the torsion of an almost complex structure can be considered as an element in Ω2,0(TM). It is easy to check that the torsion of an almost complex structure is equal to its
Higher order G-structures
Imposing
See also
Notes
- ^ Which is a Lie group mapping to the general linear group . This is often but not always a Lie subgroup; for instance, for a spin structure the map is a covering spaceonto its image.
- bifunctorin G and X.
- ^ In classical field theory, such a section describes a classical .).
- Bibcode:2005gr.qc....12115S.)
- ^ a b Besse 1987, §14.61
- ^ Kobayashi 1972
- ^ Kobayashi 1972, I.4
- ^ Gauduchon 1997
References
- Zbl 0613.53001.
- .
- Gauduchon, Paul (1997). "Canonical connections for almost-hypercomplex structures". Complex Analysis and Geometry. Pitman Research Notes in Mathematics Series. Vol. 366. Longman. pp. 123–13. ISBN 978-0-582-29276-5.
- OCLC 31374337.
- OCLC 43032711.
- Godina, Marco; Matteucci, Paolo (2003). "Reductive G-structures and Lie derivatives". S2CID 119558088.