Special and general types
A G-structure on an -manifold , for a given structure group , is a -subbundle of the frame bundle (of the tangent bundle) of .
Equivalently, this means that a -structure is a choice of reduction of the canonical structure group of the principal bundle to which the tangent bundle is associated along the given inclusion .
More generally, one can consider the case is not a subgroup but equipped with any group homomorphism . If this is instead an epimorphism one speaks of a lift of structure groups.
Both cases, in turn, can naturally be understood as special cases of twisted differential c-structures, which is a notion that applies more generally to principal infinity-bundles.
In terms of higher geometry
We give an equivalent definition of -structures in terms of higher geometry (“from the nPOV”). This serves to clarify the slightly subtle but important difference between existence and choice of -structure, and seamlessly embeds the notion into the more general context of twisted differential c-structures.
Let be a homomorphism of Lie groups. Write
for the morphism of delooping Lie groupoids ( the smooth moduli stacks of smooth - and -principal bundles, respectively).
For a smooth manifold (or generally an orbifold or Lie groupoid, etc.) Let be a -principal bundle and let be any choice of morphism representing it.
Write etc. for the hom-groupoid of smooth groupoids / smooth stacks . This is equivalently the groupoid of -principal bundles over and smooth gauge transformations between them.
Then the groupoid of -structure on (with respect to the given morphism ) is the homotopy pullback
(the groupoid of twisted c-structures).
Specifically, when is a smooth manifold of dimension , the tangent bundle canonically comes from a morphism into the moduli stack for the general linear group . Then for any group homomorphism , a -structure on is a -structure on , as above.
For the moment see at integrability of G-structures.
Relation to special holonomy
The existence of -structures on tangent bundles of Riemannian manifolds is closely related to these having special holonomy.
Let be a connected Riemannian manifold of dimension with holonomy group .
For some other subgroup, admits a torsion-free G-structure precisely if is conjugate to a subgroup of .
Moreover, the space of such -structures is the coset , where is the group of elements suchthat conjugating with them lands in .
This appears as (Joyce prop. 3.1.8)
Reduction of tangent bundle structure
For the subgroup of of matrices of positive determinant, a -structure defines an orientation.
For the orthogonal group, an -structure defines a Riemannian metric. (See the discussion at vielbein and at
For the special linear group, an -structure defines a volume form.
For the trivial group, an -structure consists of an absolute parallelism? of the manifold.
For even, a -structure defines an almost complex structure on the manifold. It must satisfy an integrability condition to be a complex structure.
Lift of tangent bundle structure
An example for a lift of structure groups is
This continues with lifts to the
Reduction of more general bundle structure
Higher geometric examples
See the list at tiwsted differential c-structure.
Need to talk about integrability conditions, and those of higher degree. Also need to discuss pseudo-groups?.
Original articles include
The relation to special holonomy is discussed in
- Dominic Joyce, Compact manifolds with special holonomy , Oxford Mathematical Monogrophs (200)
General textbooks of differential geometry include
Shoshichi Kobayashi, Katsumi Nomizu, Foundations of differential geometry
Shlomo Sternberg, Lectures on differential geometry
In higher geometry
Some discussion is in section 4.4.2 of