Construction: Splitting of the Atiyah sequence from a principal connection

Let $\pi:P\to M$ be a principal G-bundle over a smooth manifold $M$, where $G$ is a Lie group with Lie algebra $\mathfrak g$.

Write $TP$ for the tangent bundle of $P$. The right action of $G$ on $P$ induces a right action on $TP$, and the quotient bundle

is the Atiyah algebroid of $P$. The differential $d\pi:TP\to TM$ is $G$-equivariant, hence descends to a vector-bundle map (the anchor)

The vertical subbundle $\ker(d\pi)\subset TP$ identifies (via fundamental vector fields) with $P\times\mathfrak g$, and after dividing by $G$ one obtains the adjoint bundle

together with an injective bundle map $\operatorname{ad}(P)\hookrightarrow \mathrm{At}(P)$. This yields the Atiyah short exact sequence

Construction (splitting from a connection)

Let $\omega$ be a principal connection on $P$, and let $H:=\ker(\omega)\subset TP$ be its horizontal distribution. Define a bundle map

as follows. For $x\in M$ and $v_x\in T_xM$, choose any $p\in P_x:=\pi^{-1}(x)$. There is a unique horizontal vector $\widetilde v_p\in H_p$ with $d\pi(\widetilde v_p)=v_x$. Set

Well-definedness. If $p' = p\cdot g$, then horizontality is preserved by right translation and $\widetilde v_{p'} = (R_g)_*\widetilde v_p$, so $[\widetilde v_{p'}]=[\widetilde v_p]$ in $TP/G$. Hence $s_\omega$ is independent of the chosen point in the fiber.

Splitting property. By construction, $a\circ s_\omega=\mathrm{id}_{TM}$, so $s_\omega$ is a (right) splitting of the Atiyah sequence.

This construction is inverse to the correspondence in principal connections ↔ splittings of the Atiyah sequence .

Examples

1. Trivial bundle. If $P=M\times G$, then $\mathrm{At}(P)\cong TM\oplus(M\times\mathfrak g)$. A connection is determined by a $\mathfrak g$-valued $1$-form $A$ on $M$, and the splitting is

   $$s_\omega(v_x)=(v_x,\,-A_x(v_x)).$$

2. Frame bundle. If $E\to M$ is a rank-$n$ vector bundle and $P=\mathrm{Fr}(E)$ is its frame bundle , then a vector bundle connection on $E$ induces a principal connection on $P$, hence a splitting $TM\to T\mathrm{Fr}(E)/\mathrm{GL}(n)$ as above.

3. Hopf fibration. For the Hopf bundle $S^3\to S^2$ (a principal $U(1)$-bundle), the standard connection takes horizontals to be orthogonal complements of the fiber circles. The resulting $s_\omega$ identifies each tangent vector on $S^2$ with the corresponding $U(1)$-equivalence class of its horizontal lift in $TS^3$.