nLab
integrability of G-structures

Contents

Context

Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

tangent cohesion

differential cohesion

graded differential cohesion

id id fermionic bosonic bosonic Rh rheonomic reduced infinitesimal infinitesimal & étale cohesive ʃ discrete discrete continuous * \array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& ʃ &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

Contents

Idea

Given a G-structure, it is integrable or locally flat if over germs it restricts to the canonical (trivial) GG-structure. If it restricts to the canonical structure only over order-kk infinitesimal disks, then it is called order kk integrable. The obstruction to first-order integrability is the torsion of a G-structure, and hence first-order integrable GG-structures are also called torsion-free GG-structures.

Beware that some authors use the term “integrable” for “torsion-free”. This originates in concentration on the case of almost symplectic structure, i.e. GG-structure for G=Sp(2n)G = Sp(2n) the symplectic group, in which case the Darboux theorem gives that first order integrability (to symplectic structure) already implies full integrability. However, in general this is not the case. For instance for orthogonal structure, i.e. GG-structure for G=O(n)G = O(n) the orthogonal group, then the fundamental theorem of Riemannian geometry gives that the torsion obstruction to first-order integrability vanishes, exhibited by the Levi-Civita connection, but full integrability here is equivalent to this being a flat connection, which is a strong additional constraint. This is the case from which the terminology “locally flat” for “integrable” derives from.

Definition

Traditional

Let VV be a linear local model space, e.g. a vector space in plain differential geometry or super vector space in supergeometry, etc.. Write GL(V)GL(V) for its general linear group. Consider a group homomorphism GGL(V)G \longrightarrow GL(V).

Write c 0\mathbf{c}_0 for the standard flat GG-structure on VV (see at G-Structure – Examples – Standard flat G-structure).

Definition

A G-structure c\mathbf{c} on a manifold XX modeled on VV (e.g. a smooth manifold or supermanifold) is called integrable if

  1. there exists cover {U iX}\{U_i \hookrightarrow X\} by open subsets U iVU_i \hookrightarrow V;

  2. such that the GG-structure c\mathbf{c} on XX restricts on each patch to the default GG-structure c 0\mathbf{c}_0 on VV:

    c| U ic 0| U i. \mathbf{c}|_{U_i} \simeq \mathbf{c}_0|_{U_i} \,.

This is due to (Sternberg 64, section VII, def. 2.4, Guillemin 65, section 3). For review see also (Alekseevskii, Lott 90, page 4 of the exposition).

Remark

More concretely, if GG-structure is modeled by GG-subbundles PP of the frame bundle (as discussed at G-structure – In terms of subbundles of the frame bundle ), then it is integrable if each PFr(X)P \hookrightarrow Fr(X) restricts on each patch to P 0Fr(V)P_0 \hookrightarrow Fr(V)

P 0| U i Fr(V)| U i P| U i Fr(X)| U i. \array{ P_0|_{U_i} &\hookrightarrow& Fr(V)|_{U_i} \\ {}^{\mathllap{\simeq}}\downarrow && \downarrow \\ P|_{U_i} &\hookrightarrow& Fr(X)|_{U_i} } \,.
Definition

For kk \in \mathbb{N}, a G-structure c\mathbf{c} on a manifold XX modeled on VV (e.g. a smooth manifold or supermanifold) is called order-kk infinitesimally integrable if at each point xXx \in X its restriction to the order-kk infinitesimal neighbourhood 𝔻 0 V𝔻 x XX\mathbb{D}^V_0 \simeq \mathbb{D}_x^X \hookrightarrow X is equal to the default GG-structure c 0\mathbf{c}_0.

(Guillemin 65, section 4)

In higher differential cohesive geometry

One may formalize the concept of integrable GG-structure in the generality of higher differential geometry, formalized in differential cohesion. See also there at differential cohesion – G-Structure.

Let VV be framed, def. , let GG be an ∞-group and GGL(V)G \to GL(V) a homomorphism, hence

GStruc:BGBGL(V) G\mathbf{Struc}\colon \mathbf{B}G \longrightarrow \mathbf{B}GL(V)

a morphism between the deloopings.

Definition

For XX a VV-manifold, def. , a G-structure on XX is a lift of the structure group of its frame bundle, def. , to GG, hence a diagram

X BG τ X GStruc BGL(V) \array{ X && \longrightarrow&& \mathbf{B}G \\ & {}_{\mathllap{\tau_X}}\searrow && \swarrow_{\mathrlap{G\mathbf{Struc}}} \\ && \mathbf{B}GL(V) }

hence a morphism

c:τ XGStruc \mathbf{c} \colon \tau_X \longrightarrow G\mathbf{Struc}

is the slice (∞,1)-topos.

In fact GStrucH /BGL(n)G\mathbf{Struc}\in \mathbf{H}_{/\mathbf{B}GL(n)} is the moduli ∞-stack of such GG-structures.

The double slice (H /BGL(n)) /GStruc(\mathbf{H}_{/\mathbf{B}GL(n)})_{/G\mathbf{Struc}} is the (∞,1)-category of such GG-structures.

Example

If VV is framed, def. , then it carries the trivial GG-structure, which we denote by

c 0:τ VGStruc. \mathbf{c}_0 \colon \tau_{V} \longrightarrow G\mathbf{Struc} \,.
Definition

For VV framed, def. , and XX a VV-manifold, def. , then GG-structure c\mathbf{c} on XX is integrable (or locally flat) if there exists a VV-cover

U V X \array{ && U \\ & \swarrow && \searrow \\ V && && X }

such that the correspondence of frame bundles induced via remark

τ U τ V τ X \array{ && \tau_U \\ & \swarrow && \searrow \\ \tau_V && && \tau_X }

(a diagram in H /BGL(V)\mathbf{H}_{/\mathbf{B}GL(V)}) extends to a sliced correspondence between c\mathbf{c} and the trivial GG-structure c 0\mathbf{c}_0 on VV, example , hence to a diagram in H /BGL(V)\mathbf{H}_{/\mathbf{B}GL(V)} of the form

τ U τ V τ X c 0 c GStruct \array{ && \tau_U \\ & \swarrow && \searrow \\ \tau_V && \swArrow_{\mathrlap{\simeq}} && \tau_X \\ & {}_{\mathllap{\mathbf{c}_0}}\searrow && \swarrow_{\mathrlap{\mathbf{c}}} \\ && G\mathbf{Struct} }

On the other hand, c\mathbf{c} is called infinitesimally integrable (or torsion-free) if such an extension exists (only) after restriction to all infinitesimal disks in XX and UU, hence after composition with the counit

relUU \flat^{rel} U \longrightarrow U

of the relative flat modality, def. :

τ relU τ V τ X c 0 c GStruct. \array{ && \tau_{\flat^{rel} U} \\ & \swarrow && \searrow \\ \tau_V && \swArrow_{\mathrlap{\simeq}} && \tau_X \\ & {}_{\mathllap{\mathbf{c}_0}}\searrow && \swarrow_{\mathrlap{\mathbf{c}}} \\ && G\mathbf{Struct} } \,.
Remark

As before, if the given reduction modality encodes order-kk infinitesimals, then the infinitesimal integrability in def. is order-kk integrability. For k=1k = 1 this is torsion-freeness.

Properties

Existence and torsion

The obstruction for a GG-structure to be integrable to first order is its torsion of a G-structure.

In terms of adapted coordinate systems

A GG-structure on XX is integrable previsely if there exists an atlas of XX by coordinate charts with the property that their canonical frame fields are GG-frames.

(Sternberg 64, section VII, exercise 2.1)

Examples

Complex structure

A GL(n,)GL(2n,)GL(n,\mathbb{C}) \to GL(2n,\mathbb{R})-structure is an almost complex structure. Its torsion of a G-structure vanishes precisely if its Nijenhuis tensor vanishes, hence, by the Newlander-Nirenberg theorem, precisely if it is a complex structure. Since a complex manifold admits holomorphic coordinate charts, this first-order integrability already implies full integrability.

Symplectic structure

An Sp(n)GL(2n)Sp(n) \hookrightarrow GL(2n)-structure is an almost symplectic structure. Its torsion of a G-structure is the de Rham differential dω\mathbf{d}\omega of the corresponding 2-form ω\omega (recalled e.g. in Albuquerque-Picken 11). Hence first-order integrability here amounts precisely to symplectic structure. The Darboux theorem asserts that this is already a fully integrable structure.

Orthogonal structure

An O(n)GL(n)O(n)\to GL(n)-structure is an orthogonal structure, hence a vielbein, hence a Riemannian metric. The fundamental theorem of Riemannian geometry says that in this case the torsion of a G-structure vanishes, exhibited by the existence of the Levi-Civita connection. The corresponding first-order integrability is the existence of Riemann normal coordinates (since these identify the given vielbein at any point to first order with the trivial (identity) vielbein). The higher order obstructions to integrability turn out to all be proportional to combinations of the Riemann curvature. Full integrability is equivalent to the vanishing of Riemann tensor, hence to the LC-connection being a flat connection.

Unitary structure

The case of unitary structure is precisely the combination of the above three cases.

By the fact (see at unitary group – relation to orthogonal, symplectic and general linear group) that the unitary group is the intersection

U(n)O(2n)×GL(2n,)Sp(2n,)×GL(2n,)GL(n,) U(n) \simeq O(2n) \underset{GL(2n,\mathbb{R})}{\times} Sp(2n,\mathbb{R}) \underset{GL(2n,\mathbb{R})}{\times} GL(n,\mathbb{C})

a U(n)GL(2n,)U(n) \hookrightarrow GL(2n,\mathbb{R})-structure – called an almost Hermitian structure – is precisely a joint orthogonal structure, almost symplectic structure and almost complex structure. Hence if first order integrable – called a Kähler manifold structure – this is precisely a joint orthogonal structure/Riemannian manifold structure, symplectic manifold structure, complex manifold structure.

G 2G_2-Structure

A An G 2GL(7)G_2 \to GL(7)-structure is a G2-structure. Its torsion of a G-structure vanishes if the corresponding definite 3-form ω\omega is covariantly constant with respect to the induced Riemannian metric, in which case the structure is a G2-manifold. Beware that some authors refer to first-order integrable G 2G_2-structure (or even weaker conditions) as “integrable G 2G_2-structure” (see Bryant 05, remark 2 for critical discussion of the terminology). The higher-order torsion invariants of G 2G_2-structures do not in general vanish (e.g Bryant 05, (4.7)) and so, contrary to the above cases of symplectic and complex structure, G 2G_2-manifold structure does not imply integrable G 2G_2-structure.

Further examples

References

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

  • Victor Guillemin, The integrability problem for GG-structures, Trans. Amer. Math. Soc. 116 (1965), 544–560. (JSTOR)

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

Lecture notes include

  • Federica Pasquotto, Linear GG-structures by example (pdf)

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 with an eye towards special holonomy is in

  • Dominic Joyce, Compact manifolds with special holonomy, Oxford University Press 2000

See also the references at torsion of a Cartan connection and at torsion constraints in supergravity.

Last revised on January 24, 2015 at 19:16:14. See the history of this page for a list of all contributions to it.