Definition

In the setting of principal bundle morphism, a reduction of the structure group of a principal bundle P(G,π,B) over a base space B with structure group G is what follows. Suppose H is a subgroup of G, and let h:HG denote the inclusion map. A reduction of the structure group from G to H refers to the existence of a principal H-bundle P(H,π,B) and a principal bundle morphism f:PP, f=(fP,id), i.e.:

1. The diagram

PfPPππB=B

commutes, meaning πfP=π.

2. The H-equivariance condition for fP is satisfied:

fP(ph)=fP(p)h(g),pP,hH,

where h(g)=g as H is a subgroup of G.

Extension and restriction of a

Definition

([Schuller 2013] page 178. There are some gaps in the definition and in the theorem, I think. On the other hand, I think this can be also calle group reduction or group structure reduction.)

Let H be a closed subgroup of G. Let (P,π,M) a principal G-bundle and (Q,π,M) a principal H-bundle. If there exists a principal bundle morphism u from P to Q, i.e. a smooth bundle map which is equivariant with respect to the inclusion HG then Q is called an H-restriction of P and P is called a G-extension of Q.

Theorem

[Schuller 2013] page 178.

Let H be a closed Lie subgroup of G.

i) Any principal H-bundle can be extended to a principal G-bundle.

ii) A principal G-bundle (P,π,M) can be restricted to a principal H-bundle if and only if the bundle (P/H,π,M) has a section.

Proof

I don't know.

Idea of ii). Think of the orthonormal frame bundle. Also, see G-structure .

Particular cases

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Author of the notes: Antonio J. Pan-Collantes

antonio.pan@uca.es


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