nLab
G-structure

Context

Cohomology

cohomology

Special and general types

Special notions

Variants

Extra structure

Operations

Theorems

Higher geometry

Contents

Idea

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.

Definition

Traditional

(…)

In terms of higher geometry

We give an equivalent definition of GG-structures in terms of higher 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.

Definition

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 g:XBKg : X \to \mathbf{B}K be any choice of morphism representing 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){P}. \mathbf{c}Struc_{[P]}(X) := \mathbf{H}(X, \mathbf{B}G) \times_{\mathbf{H}(X, \mathbf{B}K)} \{P\} \,.
cStruc [P](X) * g H(X,BG) H(X,c) H(X,BK) \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 XX is a smooth manifold of dimension nn, the tangent bundle TXT X canonically comes from a morphism TX:XBGL(n)T X : 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 TXT X, as above.

Integrability

For the moment see at integrability of G-structures.

Properties

Relation to special holonomy

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

Theorem

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)

Examples

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?.

References

Traditional

Surveys include

Original articles include

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

Revised on September 22, 2013 17:05:18 by Urs Schreiber (89.204.139.76)