nLab
G-structure

Idea

A G-structure on an $n$-manifold $M$, for a given structure group $G$, is a $G$-subbundle of the frame bundle (of the tangent bundle) of $M$.

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

More generally, one can consider the case $G$ is not a subgroup but equipped with any group homomorphism $G\to \mathrm{GL}(n)$. 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.

Definition

Traditional

(…)

In terms of higher geometry

We give an equivalent definition of $G$-structures in terms of higher geometry (“from the nPOV”). This serves to clarify the slightly subtle but important difference between existence and choice of $G$-structure, and seamlessly embeds the notion into the more general context of twisted differential c-structures.

Definition

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

$$c:BG\to BK$$\mathbf{c} : \mathbf{B}G \to \mathbf{B}K

for the morphism of delooping Lie groupoids ( the smooth moduli stacks of smooth $K$- and $G$-principal bundles, respectively).

For $X$ a smooth manifold (or generally an orbifold or Lie groupoid, etc.) Let $P\to X$ be a $K$-principal bundle and let $g:X\to BK$ be any choice of morphism representing it.

Write $H(X,BG)$ etc. for the hom-groupoid of smooth groupoids / smooth stacks . This is equivalently the groupoid of $G$-principal bundles over $X$ and smooth gauge transformations between them.

Then the groupoid of $G$-structure on $P$ (with respect to the given morphism $G\to K$) is the homotopy pullback

$$c{\mathrm{Struc}}_{[P]}(X):=H(X,BG){\times }_{H(X,BK)}\{P\}.$$\mathbf{c}Struc_{[P]}(X) := \mathbf{H}(X, \mathbf{B}G) \times_{\mathbf{H}(X, \mathbf{B}K)} \{P\} \,.

$$\begin{array}{ccc}c{\mathrm{Struc}}_{[P]}(X)& \to & *\\ \downarrow & {⇙}_{\simeq }& {\downarrow }^g\\ H(X,BG)& \stackrel{H(X,c)}{\to }& H(X,BK)\end{array}$$\array{ \mathbf{c}Struc_{[P]}(X) &\to& * \\ \downarrow & \swArrow_\simeq& \downarrow^{g} \\ \mathbf{H}(X, \mathbf{B}G) &\stackrel{\mathbf{H}(X, \mathbf{c})}{\to}& \mathbf{H}(X, \mathbf{B}K) }

(the groupoid of twisted c-structures).

Specifically, when $X$ is a smooth manifold of dimension $n$, the tangent bundle $TX$ canonically comes from a morphism $TX:X\to B\mathrm{GL}(n)$ into the moduli stack for the general linear group $K:=\mathrm{GL}(n)$. Then for any group homomorphism $G\to \mathrm{GL}(n)$, a $G$-structure on $X$ is a $G$-structure on $TX$, as above.

Properties

Relation to special holonomy

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

Theorem

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

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

Moreover, the space of such $G$-structures is the coset $G/L$, where $L$ is the group of elements suchthat conjugating $\mathrm{Hol}(g)$ with them lands in $G$.

This appears as (Joyce prop. 3.1.8)

Examples

Reduction of tangent bundle structure

• For the subgroup of $\mathrm{GL}(n,ℝ)$ of matrices of positive determinant, a $\mathrm{GL}(n,ℝ{)}^{+}$-structure defines an orientation.

• For the orthogonal group, an $O(n)$-structure defines a Riemannian metric. (See the discussion at vielbein and at

• For the special linear group, an $\mathrm{SL}(n,R)$-structure defines a volume form.

• For the trivial group, an $\{e\}$-structure consists of an absolute parallelism? of the manifold.

• For $n=2m$ even, a $\mathrm{GL}(m,ℂ)$-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

• for the spin group $\mathrm{spin}(n)$, a $G$-structure is a spin structure.

This continues with lifts to the

• string group giving string structure;

• fivebrane group giving fivebrane structure.

Reduction of more general bundle structure

• For general $G\to K$, the corresponding notion of Cartan geometry involves $G$-structure on $K$-principal bundles (not necessarily underlying a tangent bundle).

• A $U(n,n)\hookrightarrow O(2n,2n)$-structure is a generalized complex structure;

• For $H_n\to E_{n(n)}$ the inclusion of the maximal compact subgroup into the split real form of an exceptional Lie group, the corresponding structure is an exceptional generalized geometry.

Higher geometric examples

See the list at tiwsted differential c-structure.

Further issues

Need to talk about integrability conditions, and those of higher degree. Also need to discuss pseudo-groups?.

Related concepts

• special holonomy

References

Traditional

Surveys include

• Wikipedia

Original articles include

• S.S. Chern 1966, The geometry of G-structures, Bull. Amer. Math. Soc. 72(2): 167–219.

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

• differential cohomology in a cohesive topos