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 $h : H \to G$ 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 $f : P^{'} \to P$ , $f = (f_{P}, i d)$ , i.e.:

The diagram

\begin{array}{ccc} P^{'} & \overset{f_{P}}{⟶} & P \\ π^{'} ↓ & ↓ π \\ B & = & B \end{array}

commutes, meaning $π \circ f_{P} = π^{'}$ .
2. The $H$ -equivariance condition for $f_{P}$ is satisfied:

f_{P} (p^{'} \cdot h) = f_{P} (p^{'}) \cdot h (g), \forall p^{'} \in P^{'}, h \in H,

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

Extension and restriction of a principal bundle from [Schuller 2013]

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 $H \subseteq G$ then $Q$ is called an $H$ -restriction of $P$ and $P$ is called a $G$ -extension of $Q$ .
Pasted image 20220101105109.png

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

The orthonormal frame bundle.
When the principal bundle is the frame bundle we have G-structures.