nLab G-structure





Special and general types

Special notions


Extra structure



Higher geometry



A G-structure on an nn-manifold MM, for a given structure group GG, is a GG-subbundle of the frame bundle (of the tangent bundle) of MM.

Equivalently, this means that a GG-structure is a choice of reduction of the canonical structure group GL(n) of the principal bundle to which the tangent bundle is associated along the given inclusion GGL(n)G \hookrightarrow GL(n).

More generally, one can consider the case GG is not a subgroup but equipped with any group homomorphism GGL(n)G \to GL(n). 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 this language, a GG-structure is a lift in classifying stacks (differentiable delooping stacks BG\mathbf{B}G of Lie groups GG) of the classifying map of the tangent bundle

BG Gstructure str X TX BGL(n) \array{ && \mathbf{B}G \\ & {}^{ \mathllap{G structure} }\nearrow & \big\downarrow^{ str } \\ X &\underset{ \vdash T X }{\longrightarrow}& \mathbf{B} GL(n) }

Beware the distinction to tangential structure on XX, where such a lift is considered (only) at the level of underlying classifying spaces.



Given a smooth manifold XX of dimension nn and given a Lie subgroup GGL(n)G \hookrightarrow GL(n) of the general linear group, then a GG-structure on XX is a reduction of the structure group of the frame bundle of XX to GG.

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 kkth order jet bundle. (e. g. Alekseevskii) This yields order kk GG-structure and the ordinary GG-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 are evident generalizations to supermanifolds and to complex manifolds.

In terms of subbundles of the frame bundle


Given a smooth manifold XX of dimension nn with frame bundle Fr(X)Fr(X), and given a Lie group monomorphism

GGL( n) G \longrightarrow GL(\mathbb{R}^n)

into the general linear group, then a GG-structure on XX is an GG-principal bundle PXP \to X equipped with an inclusion of fiber bundles

P Fr(X) X \array{ P &&\hookrightarrow&& Fr(X) \\ & \searrow && \swarrow \\ && X }

which is GG-equivariant.

(Sternberg 64, section VII, def. 2.1).


From this perspective, a GG-structure consists of the collection of all GG-frames on a manifold. For instance for an orthogonal structure it consists of all those frames which are pointwise an orthonormal basis of the tangent bundle (with respect to the Riemannian metric which is defined by the orthonormal structure).



Given GGL(n)G \hookrightarrow GL(n) and given any one frame field σ:XFr(X)\sigma \colon X \to Fr(X) over a manifold XX, then acting with GG on σ\sigma at each point produces a GG-subbundle. This is called the GG-structure generated by the frame field σ\sigma.

In terms of Cartan connections

A GG-structure equipped with compatible connection data is equivalently a Cartan connection for the inclusion (G nG)(G \hookrightarrow \mathbb{R}^n \rtimes G).

See at Cartan connection – Examples – G-structures

In higher differential geometry

GG-structure on a KK-principal bundle

We give an equivalent definition of GG-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 GG-structure, and seamlessly embeds the notion into the more general context of twisted differential c-structures.


Let GKG \to K be a homomorphism of Lie groups. Write

c:BGBK \mathbf{c} : \mathbf{B}G \to \mathbf{B}K

for the morphism of delooping Lie groupoids ( the smooth moduli stacks of smooth KK- and GG-principal bundles, respectively).

For XX a smooth manifold (or generally an orbifold or Lie groupoid, etc.) Let PXP \to X be a KK-principal bundle and let

k:XBK k \colon X \longrightarrow \mathbf{B}K

be any choice of morphism modulating it.

Write H(X,BG)\mathbf{H}(X, \mathbf{B}G) etc. for the hom-groupoid of smooth groupoids / smooth stacks . This is equivalently the groupoid of GG-principal bundles over XX and smooth gauge transformations between them.

Then the groupoid of GG-structure on PP (with respect to the given morphism GKG \to K) is the homotopy pullback

cStruc [P](X):=H(X,BG)× H(X,BK){k}. \mathbf{c}Struc_{[P]}(X) := \mathbf{H}(X, \mathbf{B}G) \times_{\mathbf{H}(X, \mathbf{B}K)} \{k\} \,.
cStruc [P](X) * k H(X,BG) H(X,c) H(X,BK) \array{ \mathbf{c}Struc_{[P]}(X) &\longrightarrow& \ast \\ \downarrow & \swArrow_\simeq& \downarrow^{\mathrlap{k}} \\ \mathbf{H}(X, \mathbf{B}G) &\stackrel{\mathbf{H}(X, \mathbf{c})}{\longrightarrow}& \mathbf{H}(X, \mathbf{B}K) }

(the groupoid of twisted c-structures).


If here kk is trivial in that it factors through the point, k:X*BKk \colon X \to \ast \to \mathbf{B}K then this homotopy fiber product is H(X,K/G)\mathbf{H}(X,K/G), where K/GK/G is the coset space (Klein geometry) which itself sits in the homotopy fiber sequence

K/GBGBK. K/G \to \mathbf{B}G \to \mathbf{B}K \,.

Specifically, when XX is a smooth manifold of dimension nn, the frame bundle Fr(X)Fr(X) is modulated? by a morphism τ X:XBGL(n)\tau_X \colon X \to \mathbf{B} GL(n) into the moduli stack for the general linear group K:=GL(n)K := GL(n). Then for any group homomorphism GGL(n)G \to GL(n), a GG-structure on XX is a GG-structure on Fr(X)Fr(X), as above.

GG-Structure on an etale \infty-groupoid

We discuss the concept in the generality of higher differential geometry, formalized in differential cohesion.

See at differential cohesion – G-Structure


Integrability of GG-structure


A GG-structure on a manifold XX is called locally flat (Sternberg 64, section VII, def. 24) or integrable (e.g. Alekseevskii) if it is locally equivalent to the standard flat GG-structure, def. .

This means that there is an open cover {U iX}\{U_i \to X\} by open subsets of the Cartesian space n\mathbb{R}^n such that the restriction of the GG-structure to each of these is equivalent to the standard flat GG-structure.

See at integrability of G-structures for more on this

The obstruction to integrability of GG-structure is the torsion of a G-structure. See there for more.

Relation to special holonomy

The existence of GG-structures on tangent bundles of Riemannian manifolds is closely related to these having special holonomy.


Let (X,g)(X,g) be a connected Riemannian manifold of dimension nn with holonomy group Hol(g)O(n)Hol(g) \subset O(n).

For GO(n)G \subset O(n) some other subgroup, (X,g)(X,g) admits a torsion-free G-structure precisely if Hol(g)Hol(g) is conjugate to a subgroup of GG.

Moreover, the space of such GG-structures is the coset G/LG/L, where LL is the group of elements suchthat conjugating Hol(g)Hol(g) with them lands in GG.

This appears as (Joyce prop. 3.1.8)


Canonical G-structures


For GGL(n)G \hookrightarrow GL(n) a subgroup, the standard flat GG-structure on the Cartesian space n\mathbb{R}^n is the GG-structure which is generated, via def. , from the canonical frame field on n\mathbb{R}^n (the one which is the identity at each point, under the defining identifications).


For HGH \subset G any Lie subgroup-inclusion, the canonical quotient space coprojection GG/HG \to G/H to the coset space is an HH-principal bundle that exhibits HH-structure on G/HG/H.

(e.g. Čap-Slovak 09, p. 53)

Reduction of tangent bundle 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

Complex geometric examples

  • The choice of SO(n,)SO(n, \mathbb{C}) as subgroup of GL(n,)GL(n, \mathbb{C}), determines a complex Riemannian structure;

  • CO(n,)GL(n,)CO(n, \mathbb{C}) \hookrightarrow GL(n, \mathbb{C}), a complex conformal structure;

  • Sp(2n,)GL(2n,)Sp(2n, \mathbb{C})\hookrightarrow GL(2n, \mathbb{C}), an almost symplectic structure;

  • GL(2,)GL(n,)GL(2n,),n3GL(2, \mathbb{C}) GL(n, \mathbb{C}) \hookrightarrow GL(2n, \mathbb{C}), n \geq 3, determines an almost quaternionic structure;

  • more generally a GL(m,)GL(n,)GL(m, \mathbb{C}) GL(n, \mathbb{C})-structure on a mnm n-dimensional manifold is locally identical to a Grassmannian spinor structure.

Special holonomy examples

\;normed division algebra\;𝔸\;\mathbb{A}\;\;Riemannian 𝔸\mathbb{A}-manifolds\;\;special Riemannian 𝔸\mathbb{A}-manifolds\;
\;real numbers\;\;\mathbb{R}\;\;Riemannian manifold\;\;oriented Riemannian manifold\;
\;complex numbers\;\;\mathbb{C}\;\;Kähler manifold\;\;Calabi-Yau manifold\;
\;quaternions\;\;\mathbb{H}\;\;quaternion-Kähler manifold\;\;hyperkähler manifold\;

(Leung 02)

GG-Structures on 8-manifolds

For discussion of G-structures on closed 8-manifolds see there.

Higher geometric examples

See the list at twisted differential c-structure.



The concept originates around the work of Eli Cartan (Cartan geometry) and

Textbook accounts:

See also:

Lecture notes:

Surveys include

See also

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 G-structures more generally on orbifolds:

  • A. V. Bagaev, N. I. Zhukova, The Automorphism Groups of Finite Type GG-Structures on Orbifolds, Siberian Mathematical Journal 44, 213–224 (2003) (doi:10.1023/A:1022920417785)

  • Robert Wolak, Orbifolds, geometric structures and foliations. Applications to harmonic maps, Rendiconti del seminario matematico - Universita politecnico di Torino vol. 73/1 , 3-4 (2016), 173-187 (arXiv:1605.04190)

On GG-structure for G=Br G = Br_\infty the infinite braid group:

In supergeometry

Discussion of GG-structures in supergeometry includes

In supergravity

Discussion of G-structures in supergravity and superstring theory:

In relation to torsion constraints in supergravity:

  • John Lott, The Geometry of Supergravity Torsion Constraints, Comm. Math. Phys. 133 (1990), 563–615, (exposition in arXiv:0108125)

As a way of speaking about Calabi-Yau structure and generalized Calabi-Yau structure:

In relation to BPS states/partial reduction of number of supersymmetries under KK-compactification:

and specifically so for M-theory on 8-manifolds:

See also

In complex geometry

  • Sergey Merkulov, On group theoretic aspects of the non-linear twistor transform, (pdf)

and his chapter A in

  • Yuri Manin, Gauge Field Theory and Complex Geometry, Springer.

  • Norman Wildberger, On the complexication of the classical geometries and exceptional numbers, (pdf)

  • Jun-Muk Hwang, Rational curves and prolongations of G-structures,arXiv:1703.03160

In higher geometry

Some discussion in higher differential geometry is in section 4.4.2 of

Formalization in modal homotopy type theory is in

Last revised on April 15, 2023 at 08:36:09. See the history of this page for a list of all contributions to it.