Connection on a principal bundle

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1 Definition

Let G be a Lie group with Lie algebra \frak{g} and F \to M a principal bundle for G over a smooth manifold M. A connection on F is a distribution (a subbundle of the tangent bundle) \mathcal{H} \subset TF on F, called the "horizontal distribution", which is G-invariant and complementary to the vertical distribution \mathcal{V} on F.

The decomposition TF = \mathcal{V} \oplus \mathcal{H} can be given by the projection \pi_\mathcal{V} : TF \to \mathcal{V} onto the vertical distribution. Since each vertical space \mathcal{V}_f can be identified with \frak{g} (see Principal bundle), this map \pi_V can be viewed as a \frak{g}-valued 1-form on F, a linear map \omega : TF \to \frak{g}; this is called the connection form.

The \frak{g}-valued 2-form \Omega := d\omega + [\omega,\omega] is called curvature form and measures the non-integrability of the distribution \mathcal{H}, see the theory page Connections for details.

A connection \mathcal{H} on a G-principal bundles F induces a distribution on any associated bundle E = (F \times E_o)/G (see Principal bundle) since \mathcal{H} passes trivially to F \times E_o and by G-invariance to E. The induced distribution is called a connection on E. If E_o is a vector bundle (the action of G on E_o is linear), the connection on E is closely related to a covariant derivative (see Connections).

2 Examples

A (semi-)Riemannian metric on M defines a connection on the orthonormal frame bundle F, the Levi-Civita connection: The horizontal space \mathcal{H}_f at some orthonormal basis f of T_pM consists of the derivatives of all curves f(t) in F which are parallel along their base point curve p(t) in M.

Another type of example is the canonical connection on the principal bundle G \to G/H of a reductive homogeneous space M = G/H.

For further information see [Kobayashi&Nomizu1963].


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