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.
Given a smooth manifold of dimension and given a Lie subgroup of the general linear group, then a -structure on is a reduction of the structure group of the frame bundle of to .
There are many more explicit (and more abstract) equivalent ways to say this, which we discuss below. There are also many evident variants and generalizations.
Notably one may consider reductions of the frames in the th order jet bundle. (e. g. Alekseevskii) This yields order -structure and the ordinary -structures above are then first order.
Moreover, the definition makes sense for generalized manifolds modeled on other base spaces than just Cartesian spaces. In particular there is an evident generalization to supermanifolds.
In terms of -structures
A -structure is
for each a pointed CW-complex
equipped with a pointed Serre fibration
to the classifying space (def.);
for all a pointed continuous function
which is the identity for ;
such that for all these squares commute
where the bottom map is the canonical one (def.).
The -structure is multiplicative if it is moreover equipped with a system of maps which cover the canonical multiplication maps (def.)
and which satisfy the evident associativity and unitality, for the unit, and, finally, which commute with the maps in that all these squares commute:
Similarly, an --structure is a compatible system
indexed only on the even natural numbers.
Generally, an --structure for , is a compatible system
for all , hence for all .
(Lashof 63, Stong 68, beginning of chapter II, Kochmann 96, section 1.4)
See also at B-bordism.
Examples of -structures (def. 1) include the following:
and is orthogonal structure (or “no structure”);
and the universal principal bundle-projection is framing-structure;
the classifying space of the special orthogonal group and the canonical projection is orientation structure;
the classifying space of the spin group and the canonical projection is spin structure.
Examples of --structures include
- the classifying space of the unitary group, and the canonical projection is almost complex structure.
Given a smooth manifold of dimension , and given a -structure as in def. 1, then a -structure on the manifold is an equivalence class of the following structure:
an embedding for some ;
a homotopy class of a lift of the classifying map of the tangent bundle
The equivalence relation on such structures is to be that generated by the relation if
the second inclusion factors through the first as
the lift of the classifying map factors accordingly (as homotopy classes)
In terms of subbundles of the frame bundle
(Sternberg 64, section VII, def. 2.1).
Given and given any one frame field over a manifold , then acting with on at each point produces a -subbundle. This is called the -structure generated by the frame field .
In terms of Cartan connections
A -structure equipped with compatible connection data is equivalently a Cartan connection for the inclusion .
See at Cartan connection – Examples – G-structures
In higher differential geometry
-structure on a -principal bundle
We give an equivalent definition of -structures in terms of higher differential 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 modulating 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 frame bundle is modulated? by 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.
-Structure on an etale -grouoid
We discuss the concept in the generality of higher differential geometry, formalized in differential cohesion.
See at differential cohesion – G-Structure
Integrability of -structure
See at integrability of G-structures for more on this
The obstruction to integrability of -structure is the torsion of a G-structure. See there for more.
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)
The standard flat -structure
For a subgroup, then the standard flat -structure on the Cartesian space is the -structure which is generated, via def. 4, from the canonical frame field on (the one which is the identity at each point, under the defining identifications).
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?.
The concept of topological -structure (lifts of homotopy classes of classifying maps) originates with cobordism theory. Early expositions in terms of (B,f)-structures include
The concept in differential geometry originates around the work of Eli Cartan (Cartan geometry) and
Textbook accounts include
Shlomo Sternberg, chapter VII of Lectures on differential geometry, Prentice-Hall (1964)
Shoshichi Kobayashi, Katsumi Nomizu, Foundations of differential geometry , Volume 1 (1963), Volume 2 (1969), Interscience Publishers, reprinted 1996 by Wiley Classics Library
Discussion with an eye towards special holonomy is in
- Dominic Joyce, section 2.6 of Compact manifolds with special holonomy , Oxford Mathematical Monogrophs (200)
Discussion with an eye towards torsion constraints in supergravity is in
- John Lott, The Geometry of Supergravity Torsion Constraints, Comm. Math. Phys. 133 (1990), 563–615, (exposition in arXiv:0108125)
Discussion of -structures in supergeometry includes
In higher geometry
Some discussion is in section 4.4.2 of