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

\mathrm{At}(P)\;:=\;TP/G \;\longrightarrow\; M

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)

a:\mathrm{At}(P)\longrightarrow TM.

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

\operatorname{ad}(P):=P\times_{\mathrm{Ad}}\mathfrak g,

together with an injective bundle map $\operatorname{ad}(P)\hookrightarrow \mathrm{At}(P)$ , with the negative-fundamental-vector sign convention of the linked Atiyah sequence. This yields the Atiyah short exact sequence

0\longrightarrow \operatorname{ad}(P)\longrightarrow \mathrm{At}(P)\xrightarrow{\,a\,} TM\longrightarrow 0.

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

s_\omega:TM\longrightarrow \mathrm{At}(P)

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

s_\omega(v_x)\;:=\;[\widetilde v_p]\in (TP/G)_x.

Well-definedness and splitting property

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

Trivial bundle. If $P=M\times G$ , then $\mathrm{At}(P)\cong TM\oplus(M\times\mathfrak g)$ . Using the negative vertical coordinate of the Atiyah algebroid, a connection is determined by a $\mathfrak g$ -valued $1$ -form $A$ on $M$ , and the splitting is $s_\omega(v_x)=(v_x,\,A_x(v_x)).$

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.

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 $TS^3$ .