Principal bundle of smooth manifolds

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1 Definition

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.

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.

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.

For further information, see [Kobayashi&Nomizu1963].

2 Examples

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.

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.

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.

3 References

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