Contents

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 $GL(n)$ of the principal bundle to which the tangent bundle is associated along the given inclusion $G \hookrightarrow GL(n)$.

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

Definition

General

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

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 $k$th order jet bundle. (e. g. Alekseevskii) This yields order $k$ $G$-structure and the ordinary $G$-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 spring

In terms of subbundles of the frame bundle

Definition

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

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

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

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

which is $G$-equivariant.

Remark

From this perspective, a $G$-structure consists of the collection of all $G$-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).

Accordingly:

Definition

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

In terms of Cartan connections

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

In higher differential geometry

$G$-structure on a $K$-principal bundle

We give an equivalent definition of $G$-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 $G$-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

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

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

$k \colon X \longrightarrow \mathbf{B}K$

be any choice of morphism modulating it.

Write $\mathbf{H}(X, \mathbf{B}G)$ 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

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

(the groupoid of twisted c-structures).

Remark

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

$K/G \to \mathbf{B}G \to \mathbf{B}K \,.$
Example

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

$G$-Structure on an etale $\infty$-grouoid

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

Properties

Integrability of $G$-structure

Definition

A $G$-structure on a manifold $X$ is called locally flat (Sternberg 64, section VII, def. 24) or integrable (e.g. Alekseevskii) if it is locally equivalent to the standard flat $G$-structure, def. 5.

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

See at integrability of G-structures for more on this

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

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 $Hol(g) \subset O(n)$.

For $G \subset O(n)$ some other subgroup, $(X,g)$ admits a torsion-free G-structure precisely if $Hol(g)$ 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 $Hol(g)$ with them lands in $G$.

This appears as (Joyce prop. 3.1.8)

Examples

The standard flat $G$-structure

Definition

For $G \hookrightarrow GL(n)$ a subgroup, then the standard flat $G$-structure on the Cartesian space $\mathbb{R}^n$ is the $G$-structure which is generated, via def. 2, from the canonical frame field on $\mathbb{R}^n$ (the one which is the identity at each point, under the defining identifications).

Reduction of tangent bundle structure

• For the subgroup of $GL(n, \mathbb{R})$ of matrices of positive determinant, a $GL(n, \mathbb{R})^+$-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 $SL(n,R)$-structure defines a volume form.

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

• For $n = 2 m$ even, a $GL(m, \mathbb{C})$-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 $spin(n)$, a $G$-structure is a spin structure.

This continues with lifts to the

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?.

References

The concepts originates around the work of Eli Cartan (Cartan geometry). Textbook accounts include

• Shlomo Sternberg, chapter VII of Lectures on differential geometry, Prentice-Hall (1964)

• Shoshichi Kobayashi, Katsumi Nomizu, Foundations of differential geometry

Original articles include

Surveys include

• D. V: Alekseevskii, $G$-structure on a manifold in M. Hazewinkel (ed.) Encyclopedia of Mathematics, Volume 4

• Wikipedia

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)

In supergeometry

Discussion of $G$-structures in supergeometry includes

In higher geometry

Some discussion is in section 4.4.2 of

Revised on May 30, 2015 01:07:01 by Urs Schreiber (89.204.135.46)