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)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.



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 is an evident generalization to supermanifolds.

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


A (B,f)(B,f)-structure is

  1. for each nn\in \mathbb{N} a pointed CW-complex B nTop CW */B_n \in Top_{CW}^{\ast/}

  2. equipped with a pointed Serre fibration

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

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

  3. for all n 1n 2n_1 \leq n_2 a pointed continuous function

    ι n 1,n 2:B n 1B n 2\iota_{n_1, n_2} \;\colon\; B_{n_1} \longrightarrow B_{n_2}

    which is the identity for n 1=n 2n_1 = n_2;

such that for all n 1n 2n_1 \leq n_2 \in \mathbb{N} these squares commute

B n 1 ι n 1,n 2 B n 2 f n 1 f n 2 BO(n 1) BO(n 2), \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)(B,f)-structure is multiplicative if it is moreover equipped with a system of maps μ n 1,n 2:B n 1×B n 2B n 1+n 2\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.)

B n 1×B n 2 μ n 1,n 2 B n 1+n 2 f n 1×f n 2 f n 1+n 2 BO(n 1)×BO(n 2) BO(n 1+n 2) \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=*B_0 = \ast the unit, and, finally, which commute with the maps ι\iota in that all n 1,n 2,n 3n_1,n_2, n_3 \in \mathbb{N} these squares commute:

B n 1×B n 2 id×ι n 2,n 2+n 3 B n 1×B n 2+n 3 μ n 1,n 2 μ n 1,n 2+n 3 B n 1+n 2 ι n 1+n 2,n 1+n 2+n 3 B n 1+n 2+n 3 \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} }


B n 1×B n 2 ι n 1,n 1+n 3×id B n 1+n 3×B n 2 μ n 1,n 2 μ n 1+n 3,n 2 B n 1+n 2 ι n 1+n 2,n 1+n 2+n 3 B n 1+n 2+n 3. \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 2S^2-(B,f)(B,f)-structure is a compatible system

f 2n:B 2nBO(2n) f_{2n} \colon B_{2n} \longrightarrow B O(2n)

indexed only on the even natural numbers.

Generally, an S kS^k-(B,f)(B,f)-structure for kk \in \mathbb{N}, k1k \geq 1 is a compatible system

f kn:B knBO(kn) f_{k n} \colon B_{ kn} \longrightarrow B O(k n)

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

(Lashof 63, Stong 68, beginning of chapter II, Kochmann 96, section 1.4)

See also at B-bordism.


Examples of (B,f)(B,f)-structures (def. 1) include the following:

  1. B n=BO(n)B_n = B O(n) and f n=idf_n = id is orthogonal structure (or “no structure”);

  2. B n=EO(n)B_n = E O(n) and f nf_n the universal principal bundle-projection is framing-structure;

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

  4. B n=BSpin(n)=EO(n)/Spin(n)B_n = B Spin(n) = E O(n)/Spin(n) the classifying space of the spin group and f nf_n the canonical projection is spin structure.

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

  1. B 2n=BU(n)=EO(2n)/U(n)B_{2n} = B U(n) = E O(2n)/U(n) the classifying space of the unitary group, and f 2nf_{2n} the canonical projection is almost complex structure.

Given a smooth manifold XX of dimension nn, and given a (B,f)(B,f)-structure as in def. 1, then a (B,f)(B,f)-structure on the manifold is an equivalence class of the following structure:

  1. an embedding i X:X ki_X \; \colon \; X \hookrightarrow \mathbb{R}^k for some kk \in \mathbb{N};

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

    B n g^ f n X g BO(n). \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,g^ 1)((i X) ,g^ 2)((i_{X})_1, \hat g_1) \sim ((i_{X})_,\hat g_2) if

  1. k 2k 1k_2 \geq k_1

  2. the second inclusion factors through the first as

    (i X) 2:X(i X) 1 k 1 k 2 (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)

    g^ 2:Xg^ 1B nB n. \hat g_2 \;\colon\; X \overset{\hat g_1}{\longrightarrow} B_{n} \longrightarrow B_{n} \,.

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. 7.

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)


The standard flat GG-structure


For GGL(n)G \hookrightarrow GL(n) a subgroup, then the standard flat GG-structure on the Cartesian space n\mathbb{R}^n is the GG-structure which is generated, via def. 4, from the canonical frame field on n\mathbb{R}^n (the one which is the identity at each point, under the defining identifications).

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

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?.



The concept of topological GG-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

  • D. V: Alekseevskii, GG-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 GG-structures in supergeometry includes

In higher geometry

Some discussion is in section 4.4.2 of

Revised on June 2, 2017 01:25:17 by Tim Porter (2a01:cb04:5b1:2f00:c4a0:667:289e:9942)