Equivalent descriptions of a principal connection

Let $\pi:P\to M$ be a principal G-bundle with structure group $G$. Write $V\!P=\ker(d\pi)\subset TP$ for the vertical subbundle, and note that $d\pi:TP\to \pi^*TM$ relates $TP$ to the pullback of the tangent bundle of $M$.

Theorem (TFAE: principal connection data)

The following are equivalent:

1. Principal connection. A principal connection on $P$.
2. Horizontal distribution. A smooth subbundle $H\subset TP$ such that:
   • $TP = H \oplus V\!P$ (direct sum),
   • $H$ is $G$-equivariant: $(dR_g)_p(H_p)=H_{p\cdot g}$ for all $g\in G$.
3. Equivariant splitting of $d\pi$. A $G$-equivariant bundle map $\mathrm{hor}:\pi^*TM\to TP$ such that $d\pi\circ \mathrm{hor}=\mathrm{id}_{\pi^*TM}$. Its image is the horizontal distribution $H$.
4. Connection one-form. A $\mathfrak g$-valued 1-form $\omega\in\Omega^1(P;\mathfrak g)$ such that:
   • $\omega(\xi^\#)=\xi$ for all $\xi\in\mathfrak g$ (reproduces fundamental vector fields),
   • $R_g^*\omega=\mathrm{Ad}_{g^{-1}}\omega$ for all $g\in G$. The corresponding horizontal distribution is $H=\ker\omega$.

These correspondences are inverse to each other: a connection 1-form determines horizontals via $\ker\omega$, and a horizontal distribution determines $\omega$ by projection $TP\to V\!P\cong P\times\mathfrak g$.

Examples

1. Trivial bundle. On $P=M\times G$, any $\mathfrak g$-valued 1-form $A$ on $M$ defines a principal connection with connection form $\omega = \mathrm{Ad}_{g^{-1}}A + g^{-1}dg$ in the product coordinates $(x,g)$.
2. Levi–Civita connection as a principal connection. The Levi–Civita connection on $TM$ corresponds to a principal connection on the orthonormal frame bundle $O(TM)\to M$, with horizontals defined by parallel translation of frames.
3. Connection from a splitting. If one has a $G$-equivariant splitting of $TP\to \pi^*TM$, its image gives horizontals directly, without writing a connection form.