Principal bundle of smooth manifolds

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1 Definition

A $G$${{Authors|Jost Eschenburg}}{{Stub}} == Definition== <wikitex>; A $G$-''principal bundle'' for a Lie group $G$ is a smooth bundle $\pi : F \to M$ where $G$ acts on $F$ from the right, the action is ''free'' (that is [$fg = f$ for some $f\in F$ $\Rightarrow$ $g = e$]), and the $G$-orbits are the fibres, $fG = F_p := \pi^{-1}(\pi(f))$ for every $f\in F$. Then the mapping $\phi_f : G \to F_p$, $\phi_p(g) = pg$ is a diffeomorphism which is equivariant with respect to the action of $G$ on itself by right translations. The tangent space of the fibre, $T_fF_f$ is called the ''vertical space''; by means of the differential $(d\phi_f)_e : T_eG \to T_fF_f$ it can be identified with the Lie algebra $\frak{g} = T_gG$. The vertical spaces together form an integrable distribution $\mathcal{V} \subset TF$ on $F$, the ''vertical distribution''. If $E_o$ is any $G$-space, i.e. a manifold on which $G$ acts from the left, we have a free $G$-action on $F \times E_o$ given by $g(f,x) := (fg^{-1},gx)$. Then the orbit space $E = (F \times E_o)/G$ is a bundle over $M$ with fibres diffeomorphic to $E_o$; it is called an ''associated bundle'' to the principal bundle $F$. In particular, if $E_o$ is a vector space and the $G$-action on $E_o$ is linear (a representation of $G$), then $E$ is a vector bundle over $M$, associated to the principal bundle $F$. </wikitex> == References == {{#RefList:}} [[Category:Definitions]]G$-principal bundle for a Lie group $G$$G$ is a smooth bundle $\pi : F \to M$$\pi : F \to M$ where $G$$G$ acts on $F$$F$ from the right, the action is free (that is [$fg = f$$fg = f$ for some $f\in F$$f\in F$ $\Rightarrow$$\Rightarrow$ $g = e$$g = e$]), and the $G$$G$-orbits are the fibres, $fG = F_p := \pi^{-1}(\pi(f))$$fG = F_p := \pi^{-1}(\pi(f))$ for every $f\in F$$f\in F$. Then the mapping $\phi_f : G \to F_p$$\phi_f : G \to F_p$, $\phi_p(g) = pg$$\phi_p(g) = pg$ is a diffeomorphism which is equivariant with respect to the action of $G$$G$ on itself by right translations. The tangent space of the fibre, $T_fF_f$$T_fF_f$ is called the vertical space; by means of the differential $(d\phi_f)_e : T_eG \to T_fF_f$$(d\phi_f)_e : T_eG \to T_fF_f$ it can be identified with the Lie algebra $\frak{g} = T_gG$$\frak{g} = T_gG$. The vertical spaces together form an integrable distribution $\mathcal{V} \subset TF$$\mathcal{V} \subset TF$ on $F$$F$, the vertical distribution.

If $E_o$$E_o$ is any $G$$G$-space, i.e. a manifold on which $G$$G$ acts from the left, we have a free $G$$G$-action on $F \times E_o$$F \times E_o$ given by $g(f,x) := (fg^{-1},gx)$$g(f,x) := (fg^{-1},gx)$. Then the orbit space $E = (F \times E_o)/G$$E = (F \times E_o)/G$ is a bundle over $M$$M$ with fibres diffeomorphic to $E_o$$E_o$; it is called an associated bundle to the principal bundle $F$$F$. In particular, if $E_o$$E_o$ is a vector space and the $G$$G$-action on $E_o$$E_o$ is linear (a representation of $G$$G$), then $E$$E$ is a vector bundle over $M$$M$, associated to the principal bundle $F$$F$.