Principal bundle of smooth manifolds

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1 Definition

An $H$$\newcommand{\Z}{\mathbb{Z}} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\N}{\mathbb{N}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\F}{\mathbb{F}} \newcommand{\bZ}{\mathbb{Z}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bH}{\mathbb{H}} \newcommand{\bQ}{\mathbb{Q}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bN}{\mathbb{N}} \DeclareMathOperator\id{id} % identity map \DeclareMathOperator\Sq{Sq} % Steenrod squares \DeclareMathOperator\Homeo{Homeo} % group of homeomorphisms of a topoloical space \DeclareMathOperator\Diff{Diff} % group of diffeomorphisms of a smooth manifold \DeclareMathOperator\SDiff{SDiff} % diffeomorphism under some constraint \DeclareMathOperator\Hom{Hom} % homomrphism group \DeclareMathOperator\End{End} % endomorphism group \DeclareMathOperator\Aut{Aut} % automorphism group \DeclareMathOperator\Inn{Inn} % inner automorphisms \DeclareMathOperator\Out{Out} % outer automorphism group \DeclareMathOperator\vol{vol} % volume \newcommand{\GL}{\text{GL}} % general linear group \newcommand{\SL}{\text{SL}} % special linear group \newcommand{\SO}{\text{SO}} % special orthogonal group \newcommand{\O}{\text{O}} % orthogonal group \newcommand{\SU}{\text{SU}} % special unitary group \newcommand{\Spin}{\text{Spin}} % Spin group \newcommand{\RP}{\Rr\mathrm P} % real projective space \newcommand{\CP}{\Cc\mathrm P} % complex projective space \newcommand{\HP}{\Hh\mathrm P} % quaternionic projective space \newcommand{\Top}{\mathrm{Top}} % topological category \newcommand{\PL}{\mathrm{PL}} % piecewise linear category \newcommand{\Cat}{\mathrm{Cat}} % any category \newcommand{\KS}{\text{KS}} % Kirby-Siebenmann class \newcommand{\Hud}{\text{Hud}} % Hudson torus \newcommand{\Ker}{\text{Ker}} % Kernel \newcommand{\underbar}{\underline} %Classifying Spaces for Families of Subgroups \newcommand{\textup}{\text} \newcommand{\sp}{^}H$-principal bundle for a Lie group $H$$H$ is a smooth bundle $\pi : F \to M$$\pi : F \to M$ where $H$$H$ acts on $F$$F$ from the right, the action is free (that is if $fh = f$$fh = f$ for some $f\in F$$f\in F$ then $h = e$$h = e$), and the $H$$H$-orbits are precisely the fibres, $fH = F_p := \pi^{-1}(\pi(f))$$fH = F_p := \pi^{-1}(\pi(f))$ for every $f\in F$$f\in F$ with $p = \pi(f)$$p = \pi(f)$. The group $H$$H$ is called the structure group of the principal bundle $F$$F$.

The mapping $\phi_f : H \to F_p$$\phi_f : H \to F_p$, $\phi_p(h) = ph$$\phi_p(h) = ph$ is a diffeomorphism which is $H$$H$ equivariant where $H$$H$ acts on itself by right translations. The tangent space of the fibre, $T_fF_f$$T_fF_f$ is called vertical space; by means of the differential $(d\phi_f)_e : T_eH \to T_fF_f$$(d\phi_f)_e : T_eH \to T_fF_f$ it can be identified with the Lie algebra $\mathfrak{h} = T_eH$$\mathfrak{h} = T_eH$. The vertical spaces together form an integrable distribution $\mathcal{V} \subset TF$$\mathcal{V} \subset TF$ on $F$$F$, called vertical distribution.

If $E_o$$E_o$ is any $H$$H$-space, i.e. a manifold on which $H$$H$ acts from the left, we have a free $H$$H$-action on $F\times E_o$$F\times E_o$ given by $(f,x) := (fh^{-1},hx)$$(f,x) := (fh^{-1},hx)$. Then the orbit space $E = (F\times E_o)/H$$E = (F\times E_o)/H$ is a bundle over $M$$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$$E_o$ is a vector space and the $H$$H$-action on $E_o$$E_o$ is linear (a representation of $H$$H$), then $E$$E$ is a vector bundle over $M$$M$, associated to the principal bundle $F$$F$.

For further information, see [Kobayashi&Nomizu1963].

2 Examples

A main example is the frame bundle $F = FM$$F = FM$ of a manifold $M$$M$ whose fibre over $p\in M$$p\in M$ is the set of all bases of the tangent space $T_pM$$T_pM$. The group $GL_n$$GL_n$ acts on $F$$F$ as follows: $A = (a_{ij})\in GL_n$$A = (a_{ij})\in GL_n$ is sending a basis $b = (b_1,\dots, b_n) \in F_p$$b = (b_1,\dots, b_n) \in F_p$ onto the basis $bA = (\sum_i b_ia_{i1},\dots,\sum_i b_ia_{in})\in F_p$$bA = (\sum_i b_ia_{i1},\dots,\sum_i b_ia_{in})\in F_p$. Moreover, if $M$$M$ is equipped with a Riemannian metric, there is the orthogonal frame bundle $F = OFM$$F = OFM$ where $F_p$$F_p$ consists of the set of orthonormal bases on $T_pM$$T_pM$; this is acted on by the orthogonal group $O_n$$O_n$ in a similar way. If $M$$M$ is a Kähler manifold (a Riemannian manifold with a parallel and orthogonal almost complex structure $J$$J$ on its tangent bundle), we have the principal bundle of unitary frames (orthonormal frames of type $(b_1,Jb_1,\dots,b_m,Jb_m)$$(b_1,Jb_1,\dots,b_m,Jb_m)$) with structure group $U_m$$U_m$.

A different type of examples comes from homogeneous spaces. If $M$$M$ is a manifold and $G$$G$ a Lie group acting transitively on $M$$M$ by diffeomorphisms, $G$$G$ is a principal bundle over $M$$M$ in various ways: Fixing $p\in M$$p\in M$ we have the bundle $\pi : G \to M$$\pi : G \to M$, $g \mapsto gp$$g \mapsto gp$. Its fibre over $p$$p$ is the isotropy group $H = \{g\in G: gp = p\}$$H = \{g\in G: gp = p\}$, while the fibre over $gp$$gp$ is $gH$$gH$. Thus $G$$G$ is an $H$$H$-principal bundle where $H$$H$ acts on $G$$G$ by right multiplication. When we identify $M$$M$ with the coset space $G/H = \{gH: g\in G\}$$G/H = \{gH: g\in G\}$ using the $G$$G$-equivariant map $gH \mapsto gp$$gH \mapsto gp$, the principal bundle $\pi : G \to M$$\pi : G \to M$ is just the canonical projection $G \to G/H$$G \to G/H$.

We may embed the principal bundle $G\to M$$G\to M$ into the frame bundle $FM \to M$$FM \to M$ as follows. Fixing any basis $b = (b_1,\dots,b_n)$$b = (b_1,\dots,b_n)$ of $T_pM$$T_pM$ for some $p\in M$$p\in M$, we map $g\in G$$g\in G$ (viewed as a diffeomorphism on $M$$M$) onto the basis $b' = gb = (dg_pb_1,\dots,dg_pb_n)$$b' = gb = (dg_pb_1,\dots,dg_pb_n)$ of $T_{gp}M$$T_{gp}M$. Thus the structure group $H$$H$ of $G\to M$$G\to M$ becomes a subgroup of $GL_n$$GL_n$, the structure group of $FM$$FM$.

3 References

• [Kobayashi&Nomizu1963] S. Kobayashi and K. Nomizu, Foundations of differential geometry. Vol I, Interscience Publishers, a division of John Wiley & Sons, New York-London, 1963. MR1393940 (97c:53001a) Zbl 0508.53002