Splitting of the Atiyah sequence

Let $\pi:P\to M$ be a principal $G$-bundle with Atiyah sequence

as in the Atiyah sequence .

A splitting of the Atiyah sequence is a vector bundle map

such that $a\circ \sigma=\mathrm{id}_{TM}$. Equivalently, $\sigma$ is a right inverse of the anchor map.

Splittings are in natural bijection with principal connections on $P$:

• Given a principal connection with horizontal distribution $H\subset TP$, define $\sigma(v_x)$ to be the class in $TP/G$ of the horizontal lift $v^H_p\in H_p$ at any $p\in P_x$ (well-defined because horizontality is $G$-invariant).
• Given a splitting $\sigma$, define $H_p\subset T_pP$ by declaring $X\in T_pP$ to be horizontal iff its class $[X]\in (TP/G)_x$ equals $\sigma(d\pi_p X)$. This yields a $G$-invariant complement to the vertical space.

The curvature of the induced connection can be recovered from $\sigma$ as the obstruction to $\sigma$ preserving brackets in the Atiyah algebroid: the $\mathrm{ad}(P)$-valued 2-form

vanishes exactly when the connection is flat.

Examples

1. Trivial bundle and a gauge potential. For $P=M\times G$, identify $TP/G\cong TM\oplus (M\times \mathfrak{g})$. A choice of $\mathfrak{g}$-valued 1-form $A$ gives a splitting $\sigma(v)=(v,-A(v))$, whose associated principal connection has local connection form $A$.

2. Flatness as a bracket condition. In the trivial-bundle identification above, the splitting coming from $A$ preserves the Lie algebroid bracket precisely when $F=dA+\tfrac12[A\wedge A]=0$, i.e. when the connection is flat.

3. Circle bundles. For a principal $U(1)$-bundle, any choice of connection 1-form determines a splitting. Since $\mathfrak{u}(1)$ is abelian, the curvature is an ordinary 2-form measuring the failure of horizontal lifts to commute.