Principal bundle of smooth manifolds
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− | == Definition== | + | == Definition == |
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− | + | An $H$-''principal bundle'' for a Lie group $H$ is a smooth bundle $\pi : F \to M$ where $H$ acts on $F$ from the right, the action is ''free'' (that is if $fh = f$ for some $f\in F$ then $h = e$), and the $H$-orbits are precisely the fibres, $fH = F_p := \pi^{-1}(\pi(f))$ for every $f\in F$ with $p = \pi(f)$. The group $H$ is called the ''structure group'' of the principal bundle $F$. | |
− | $\pi : F \to M$ where $ | + | |
− | ''free'' (that is | + | |
− | $f\in F$ $ | + | |
− | $ | + | |
− | every $f\in F$ | + | |
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− | $ | + | |
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− | If $E_o$ is any $ | + | The mapping $\phi_f : H \to F_p$, $\phi_p(h) = ph$ is a diffeomorphism which is $H$ equivariant where $H$ acts on itself by right translations. The tangent space of the fibre, $T_fF_f$ is called ''vertical space''; by means of the differential $(d\phi_f)_e : T_eH \to T_fF_f$ it can be identified with the Lie algebra $\mathfrak{h} = T_eH$. The vertical spaces together form an integrable distribution $\mathcal{V} \subset TF$ on $F$, called ''vertical distribution''. |
− | we have a free $ | + | |
− | Then the orbit space $E = (F \times E_o)/ | + | If $E_o$ is any $H$-space, i.e. a manifold on which $H$ acts from the left, we have a free $H$-action on $F\times E_o$ given by $(f,x) := (fh^{-1},hx)$. Then the orbit space $E = (F\times E_o)/H$ 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 $H$-action on $E_o$ is linear (a representation of $H$), then $E$ is a vector bundle over $M$, associated to the principal bundle $F$. |
− | diffeomorphic to $E_o$; it is called an ''associated bundle'' to the | + | |
− | principal bundle $F$. | + | For further information, see \cite{Kobayashi&Nomizu1963}. |
− | In particular, if $E_o$ is a vector space and the $ | + | </wikitex> |
− | (a representation of $ | + | |
− | then $E$ is a vector bundle over $M$, associated to the principal bundle $F$. | + | == Examples == |
+ | <wikitex>; | ||
+ | A main example is the ''frame bundle'' $F = FM$ of a manifold $M$ whose fibre over $p\in M$ is the set of all bases of the tangent space $T_pM$. The group $GL_n$ acts on $F$ as follows: $A = (a_{ij})\in GL_n$ is sending a basis $b = (b_1,\dots, b_n) \in F_p$ onto the basis $bA = (\sum_i b_ia_{i1},\dots,\sum_i b_ia_{in})\in F_p$. Moreover, if $M$ is equipped with a Riemannian metric, there is the ''orthogonal frame bundle'' $F = OFM$ where $F_p$ consists of the set of orthonormal bases on $T_pM$; this is acted on by the orthogonal group $O_n$ in a similar way. If $M$ is a ''Kähler manifold'' (a Riemannian manifold with a parallel and orthogonal almost complex structure $J$ on its tangent bundle), we have the principal bundle of ''unitary frames'' (orthonormal frames of type $(b_1,Jb_1,\dots,b_m,Jb_m)$) with structure group $U_m$. | ||
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+ | A different type of examples comes from ''homogeneous spaces''. If $M$ is a manifold and $G$ a Lie group acting transitively on $M$ by diffeomorphisms, $G$ is a principal bundle over $M$ in various ways: Fixing $p\in M$ we have the bundle $\pi : G \to M$, $g \mapsto gp$. Its fibre over $p$ is the isotropy group $H = \{g\in G: gp = p\}$, while the fibre over $gp$ is $gH$. Thus $G$ is an $H$-principal bundle where $H$ acts on $G$ by right multiplication. When we identify $M$ with the coset space $G/H = \{gH: g\in G\}$ using the $G$-equivariant map $gH \mapsto gp$, the principal bundle $\pi : G \to M$ is just the canonical projection $G \to G/H$. | ||
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+ | We may embed the principal bundle $G\to M$ into the frame bundle $FM \to M$ as follows. Fixing any basis $b = (b_1,\dots,b_n)$ of $T_pM$ for some $p\in M$, we map $g\in G$ (viewed as a diffeomorphism on $M$) onto the basis $b' = gb = (dg_pb_1,\dots,dg_pb_n)$ of $T_{gp}M$. Thus the structure group $H$ of $G\to M$ becomes a subgroup of $GL_n$, the structure group of $FM$. | ||
</wikitex> | </wikitex> | ||
== References == | == References == | ||
{{#RefList:}} | {{#RefList:}} | ||
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[[Category:Definitions]] | [[Category:Definitions]] | ||
+ | [[Category:Connections and curvature]] |
Latest revision as of 16:06, 15 May 2013
The user responsible for this page is Jost Eschenburg. No other user may edit this page at present. |
This page has not been refereed. The information given here might be incomplete or provisional. |
1 Definition
An -principal bundle for a Lie group is a smooth bundle where acts on from the right, the action is free (that is if for some then ), and the -orbits are precisely the fibres, for every with . The group is called the structure group of the principal bundle .
The mapping , is a diffeomorphism which is equivariant where acts on itself by right translations. The tangent space of the fibre, is called vertical space; by means of the differential it can be identified with the Lie algebra . The vertical spaces together form an integrable distribution on , called vertical distribution.
If is any -space, i.e. a manifold on which acts from the left, we have a free -action on given by . Then the orbit space is a bundle over with fibres diffeomorphic to ; it is called an associated bundle to the principal bundle . In particular, if is a vector space and the -action on is linear (a representation of ), then is a vector bundle over , associated to the principal bundle .
For further information, see [Kobayashi&Nomizu1963].
2 Examples
A main example is the frame bundle of a manifold whose fibre over is the set of all bases of the tangent space . The group acts on as follows: is sending a basis onto the basis . Moreover, if is equipped with a Riemannian metric, there is the orthogonal frame bundle where consists of the set of orthonormal bases on ; this is acted on by the orthogonal group in a similar way. If is a Kähler manifold (a Riemannian manifold with a parallel and orthogonal almost complex structure on its tangent bundle), we have the principal bundle of unitary frames (orthonormal frames of type ) with structure group .
A different type of examples comes from homogeneous spaces. If is a manifold and a Lie group acting transitively on by diffeomorphisms, is a principal bundle over in various ways: Fixing we have the bundle , . Its fibre over is the isotropy group , while the fibre over is . Thus is an -principal bundle where acts on by right multiplication. When we identify with the coset space using the -equivariant map , the principal bundle is just the canonical projection .
We may embed the principal bundle into the frame bundle as follows. Fixing any basis of for some , we map (viewed as a diffeomorphism on ) onto the basis of . Thus the structure group of becomes a subgroup of , the structure group of .
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