Contents

cohomology

# 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 are evident generalizations to supermanifolds and to complex manifolds.

### In terms of $(B,f)$-structures

###### Definition

A $(B,f)$-structure is

1. for each $n\in \mathbb{N}$ a pointed CW-complex $B_n \in Top_{CW}^{\ast/}$

2. equipped with a pointed Serre fibration

$\array{ B_n \\ \downarrow^{\mathrlap{f_n}} \\ B O(n) }$

to the classifying space $B O(n)$ (def.);

3. for all $n_1 \leq n_2$ a pointed continuous function

$\iota_{n_1, n_2} \;\colon\; B_{n_1} \longrightarrow B_{n_2}$

which is the identity for $n_1 = n_2$;

such that for all $n_1 \leq n_2 \in \mathbb{N}$ these squares commute

$\array{ B_{n_1} &\overset{\iota_{n_1,n_2}}{\longrightarrow}& B_{n_2} \\ {}^{\mathllap{f_{n_1}}}\downarrow && \downarrow^{\mathrlap{f_{n_2}}} \\ B O(n_1) &\longrightarrow& B O(n_2) } \,,$

where the bottom map is the canonical one (def.).

The $(B,f)$-structure is multiplicative if it is moreover equipped with a system of maps $\mu_{n_1,n_2} \colon B_{n_1}\times B_{n_2} \to B_{n_1 + n_2}$ which cover the canonical multiplication maps (def.)

$\array{ B_{n_1} \times B_{n_2} &\overset{\mu_{n_1, n_2}}{\longrightarrow}& B_{n_1 + n_2} \\ {}^{\mathllap{f_{n_1} \times f_{n_2}}}\downarrow && \downarrow^{\mathrlap{f_{n_1 + n_2}}} \\ B O(n_1) \times B O(n_2) &\longrightarrow& B O(n_1 + n_2) }$

and which satisfy the evident associativity and unitality, for $B_0 = \ast$ the unit, and, finally, which commute with the maps $\iota$ in that all $n_1,n_2, n_3 \in \mathbb{N}$ these squares commute:

$\array{ B_{n_1} \times B_{n_2} &\overset{id \times \iota_{n_2,n_2+n_3}}{\longrightarrow}& B_{n_1} \times B_{n_2 + n_3} \\ {}^{\mathllap{\mu_{n_1, n_2}}}\downarrow && \downarrow^{\mathrlap{\mu_{n_1,n_2 + n_3}}} \\ B_{n_1 + n_2} &\underset{\iota_{n_1+n_2, n_1 + n_2 + n_3}}{\longrightarrow}& B_{n_1 + n_2 + n_3} }$

and

$\array{ B_{n_1} \times B_{n_2} &\overset{\iota_{n_1,n_1+n_3} \times id}{\longrightarrow}& B_{n_1+n_3} \times B_{n_2 } \\ {}^{\mathllap{\mu_{n_1, n_2}}}\downarrow && \downarrow^{\mathrlap{\mu_{n_1 + n_3 , n_2}}} \\ B_{n_1 + n_2} &\underset{\iota_{n_1+n_2, n_1 + n_2 + n_3}}{\longrightarrow}& B_{n_1 + n_2 + n_3} } \,.$

Similarly, an $S^2$-$(B,f)$-structure is a compatible system

$f_{2n} \colon B_{2n} \longrightarrow B O(2n)$

indexed only on the even natural numbers.

Generally, an $S^k$-$(B,f)$-structure for $k \in \mathbb{N}$, $k \geq 1$ is a compatible system

$f_{k n} \colon B_{ kn} \longrightarrow B O(k n)$

for all $n \in \mathbb{N}$, hence for all $k n \in k \mathbb{N}$.

###### Example

Examples of $(B,f)$-structures (def. ) include the following:

1. $B_n = B O(n)$ and $f_n = id$ is orthogonal structure (or “no structure”);

2. $B_n = E O(n)$ and $f_n$ the universal principal bundle-projection is framing-structure;

3. $B_n = B SO(n) = E O(n)/SO(n)$ the classifying space of the special orthogonal group and $f_n$ the canonical projection is orientation structure;

4. $B_n = B Spin(n) = E O(n)/Spin(n)$ the classifying space of the spin group and $f_n$ the canonical projection is spin structure.

Examples of $S^2$-$(B,f)$-structures include

1. $B_{2n} = B U(n) = E O(2n)/U(n)$ the classifying space of the unitary group, and $f_{2n}$ the canonical projection is almost complex structure.
###### Definition

Given a smooth manifold $X$ of dimension $n$, and given a $(B,f)$-structure as in def. , then a $(B,f)$-structure on the manifold is an equivalence class of the following structure:

1. an embedding $i_X \; \colon \; X \hookrightarrow \mathbb{R}^k$ for some $k \in \mathbb{N}$;

2. a homotopy class of a lift $\hat g$ of the classifying map $g$ of the tangent bundle

$\array{ && B_{n} \\ &{}^{\mathllap{\hat g}}\nearrow& \downarrow^{\mathrlap{f_n}} \\ X &\overset{g}{\hookrightarrow}& B O(n) } \,.$

The equivalence relation on such structures is to be that generated by the relation $((i_{X})_1, \hat g_1) \sim ((i_{X})_,\hat g_2)$ if

1. $k_2 \geq k_1$

2. the second inclusion factors through the first as

$(i_X)_2 \;\colon\; X \overset{(i_X)_1}{\hookrightarrow} \mathbb{R}^{k_1} \hookrightarrow \mathbb{R}^{k_2}$
3. the lift of the classifying map factors accordingly (as homotopy classes)

$\hat g_2 \;\colon\; X \overset{\hat g_1}{\longrightarrow} B_{n} \longrightarrow B_{n} \,.$

### 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 $\infty$-groupoid

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. .

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. , 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

### Complex geometric examples

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

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

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

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

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

### Special holonomy examples

$\;$normed division algebra$\;$$\;\mathbb{A}\;$$\;$Riemannian $\mathbb{A}$-manifolds$\;$$\;$special Riemannian $\mathbb{A}$-manifolds$\;$
$\;$real numbers$\;$$\;\mathbb{R}\;$$\;$Riemannian manifold$\;$$\;$oriented Riemannian manifold$\;$
$\;$complex numbers$\;$$\;\mathbb{C}\;$$\;$Kähler manifold$\;$$\;$Calabi-Yau manifold$\;$
$\;$quaternions$\;$$\;\mathbb{H}\;$$\;$quaternion-Kähler manifold$\;$$\;$hyperkähler manifold$\;$
$\;$octonions$\;$$\;\mathbb{O}\;$$\;$Spin(7)-manifold$\;$$\;$G2-manifold$\;$

(Leung 02)

### $G$-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 of topological $G$-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

Surveys include

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

and specifically in supergravity:

See also at torsion constraints in supergravity.

### 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