Induced covariant derivative on sections of an associated vector bundle

A smooth section $s\in\Gamma(E)$ can be represented by a smooth $\rho$-equivariant map $f:P\to V$ satisfying

Given a vector field $X$ on $M$ and a point $x\in M$, pick $p\in P_x$ and let $\widetilde{X}_p\in H_p$ be the horizontal lift of $X_x$ at $p$. Define

Why this is well-defined

• If $p$ is replaced by $p\cdot g$, then $\widetilde{X}_{p\cdot g} = (dR_g)_p(\widetilde{X}_p)$ because the horizontal distribution is $G$-invariant.
• Using the equivariance $f(p\cdot g)=\rho(g^{-1})f(p)$ and differentiating along a horizontal direction shows that $df_{p\cdot g}(\widetilde{X}_{p\cdot g})$ represents the same element of the fiber quotient $P\times_G V$.

The operator $s\mapsto \nabla s$ is a connection on the vector bundle $E$, and it satisfies the Leibniz rule (see the induced connection satisfies the Leibniz rule). In particular, this construction is the section-level version of inducing connections on associated bundles using horizontals.

Local formula

Choose a local section $s_0:U\to P$. Writing a section of $E$ as $s(x)=[s_0(x),v(x)]$ with $v:U\to V$, let

be the local connection 1-form. Let $\rho_*:\mathfrak g\to \mathrm{End}(V)$ be the induced Lie algebra representation (obtained by differentiating $\rho$, as in differentiating a Lie group map). Then locally,

1. Trivial bundle with a matrix-valued potential. For $P=U\times G$ and $E=U\times V$, a choice of $A\in\Omega^1(U;\mathfrak g)$ produces
   $$\nabla = d + \rho_*(A).$$
   If $A=0$ this reduces to the ordinary derivative and corresponds to the flat connection on a trivial bundle.

1. Adjoint bundle. Take $V=\mathfrak g$ with $\rho=\mathrm{Ad}$. Then $E$ is the adjoint bundle $\mathrm{Ad}(P)$. In a local trivialization with local connection form $A$, the induced covariant derivative on a section $\phi$ of $\mathrm{Ad}(P)$ has the familiar form
   $$\nabla \phi = d\phi + [A,\phi],$$
   which underlies the covariant exterior derivative on adjoint-valued forms.

1. Tangent bundle from the frame bundle. Let $P=\mathrm{Fr}(TM)$ be the frame bundle of a manifold and let $V=\mathbb R^n$ with the standard representation of $\mathrm{GL}(n,\mathbb R)$. Then $TM\cong P\times_{\mathrm{GL}(n)}\mathbb R^n$, and a principal connection on $P$ induces the usual covariant derivative on vector fields. In the Riemannian case, choosing the orthonormal frame bundle and its connection recovers the Levi-Civita connection.