Connection Form - Structure Groups

Structure Groups

A more specific type of connection form can be constructed when the vector bundle E carries a structure group. This amounts to a preferred class of frames e on E, which are related by a Lie group G. For example, in the presence of a metric in E, one works with frames that form an orthonormal basis at each point. The structure group is then the orthogonal group, since this group preserves the orthonormality of frames. Other examples include:

• The usual frames, considered in the preceding section, have structural group GL(k) where k is the fibre dimension of E.
• The holomorphic tangent bundle of a complex manifold (or almost complex manifold). Here the structure group is GL_n(C) ⊂ GL_2n(R). In case a hermitian metric is given, then the structure group reduces to the unitary group acting on unitary frames.
• Spinors on a manifold equipped with a spin structure. The frames are unitary with respect to an invariant inner product on the spin space, and the group reduces to the spin group.
• Holomorphic tangent bundles on CR manifolds.

In general, let E be a given vector bundle of fibre dimension k and G ⊂ GL(k) a given Lie subgroup of the general linear group of Rk. If (e_α) is a local frame of E, then a matrix-valued function (g_ij): M → G may act on the e_α to produce a new frame

Two such frames are G-related. Informally, the vector bundle E has the structure of a G-bundle if a preferred class of frames is specified, all of which are locally G-related to each other. In formal terms, E is a fibre bundle with structure group G whose typical fibre is Rk with the natural action of G as a subgroup of GL(k).