Theorem: Principal connections are equivalent to splittings of the Atiyah sequence

Let $\pi:P\to M$ be a principal G-bundle with structure Lie group $G$ and Lie algebra $\mathfrak g$.

The Atiyah sequence is the short exact sequence of vector bundles over $M$

where $a$ is induced by $d\pi$.

Theorem

There is a natural bijection between:

• principal connections on $P$, and
• vector bundle maps $s:TM\to TP/G$ such that $a\circ s=\mathrm{id}_{TM}$ (i.e. splittings of the Atiyah sequence).

Under this bijection:

• Given a principal connection with horizontal distribution $H\subset TP$, the splitting is the map $s_\omega$ obtained by taking horizontal lifts and passing to $TP/G$ (constructed in splitting from a principal connection ).

• Given a splitting $s$, the corresponding horizontal distribution at $p\in P_x$ is the unique subspace $H_p\subset T_pP$ projecting isomorphically to $T_xM$ and representing the class $s(T_xM)\subset (TP/G)_x$; $G$-equivariance of $H$ follows from the definition of $TP/G$.

Examples

1. Trivial bundle. For $P=M\times G$, the Atiyah algebroid is $TP/G\cong TM\oplus(M\times\mathfrak g)$. A splitting is a bundle map $TM\to TM\oplus(M\times\mathfrak g)$ of the form $v\mapsto (v,-A(v))$, hence corresponds to a $\mathfrak g$-valued $1$-form $A$, i.e. a connection.

2. Frame bundle. For $P=\mathrm{Fr}(E)$, splittings of the Atiyah sequence correspond to covariant derivatives on $E$ via the equivalence between principal connections on $\mathrm{Fr}(E)$ and vector bundle connections.

3. Geometry via horizontals. Interpreting a splitting as horizontals in $TP$ makes the existence/uniqueness of horizontal lifts and the construction of parallel transport immediate consequences of basic ODE theory on the principal bundle.