Associated connection theorem

Let $\pi:P\to M$ be a principal G-bundle with structure group $G$, and let $\omega$ be a principal connection on $P$ (equivalently a horizontal distribution; see horizontal distribution and connection 1-form ).

Let $F$ be a smooth left $G$-space (see smooth G-action ) and form the associated bundle

as in the standard associated bundle construction .

Theorem (Induced connection on an associated bundle)

The principal connection $\omega$ induces a unique Ehresmann connection on $E\to M$ characterized by either of the following equivalent descriptions:

1. (Horizontal lift description)
   For each $p\in P$ and $X\in T_{\pi(p)}M$, let $\widetilde X\in T_pP$ be the $\omega$-horizontal lift (see horizontal lift of a tangent vector ). For $[p,f]\in E$, define the horizontal subspace

   $$H_{[p,f]}E := (d q)_{(p,f)}\big(H_pP \times \{0\}\big),$$

   where $q:P\times F\to P\times_G F$ is the quotient map. This gives a smooth horizontal distribution complementary to the vertical bundle of $E$.

2. (Parallel transport description)
   The principal connection defines parallel transport in $P$ (see parallel transport along a curve and parallel transport ). Transporting $(p,f)\in P\times F$ by transporting $p$ horizontally and keeping $f$ fixed produces a well-defined transport in $E$, which is exactly the induced connection.

If $F=V$ is a vector space and the $G$-action comes from a representation $\rho:G\to \mathrm{GL}(V)$, then $E=P\times_\rho V$ is an associated vector bundle , and the induced Ehresmann connection is equivalent to a vector bundle connection $\nabla$ on $E$ (compare connection on a vector bundle ). One concrete formulation is that $\nabla$ acts on sections via the covariant derivative obtained from horizontal lifts, as in the induced covariant derivative on associated sections .

Moreover, the curvature of the induced connection is obtained by applying the representation to the principal curvature (see principal curvature and curvature of induced associated connections ).

Examples

1. Levi-Civita connection induces the usual covariant derivative on tensor bundles.
   On a Riemannian manifold, the orthonormal frame bundle $O(M)\to M$ (see orthonormal frame bundle ) carries the Levi-Civita connection . Any tensor bundle built from the standard representation of $O(n)$ (e.g. $TM$, $T^*M$, $\Lambda^kT^*M$) is an associated vector bundle, and the theorem recovers the usual Levi-Civita covariant derivative on those bundles.

2. Adjoint bundle connection.
   Taking $F=\mathfrak g$ with the adjoint action, the associated bundle is $\mathrm{ad}(P)=P\times_{\mathrm{Ad}}\mathfrak g$ (see adjoint bundle ). The induced connection is the standard “adjoint bundle connection,” and the associated covariant exterior derivative on $\mathfrak g$-valued tensorial forms is exactly the operator described in covariant exterior derivative on adjoint-valued forms .

3. Dirac monopole as an induced connection on a line bundle.
   The Hopf fibration $S^3\to S^2$ is a principal $U(1)$-bundle (see Hopf fibration ). A principal connection on it (for instance the Dirac monopole connection ) induces a connection on every associated complex line bundle $S^3\times_{U(1)}\mathbb C$, recovering the usual covariant derivative used in the monopole picture.