Horizontal subbundle

Let $\pi:E\to M$ be a surjective submersion, and let

be the vertical subbundle.

Definition. A horizontal subbundle is a smooth subbundle $HE\subset \,TE$ such that for every $e\in E$,

Equivalently, $TE=HE\oplus VE$ as vector bundles over $E$.

A choice of horizontal subbundle is exactly the same data as an Ehresmann connection . In particular, the restriction of $d\pi_e$ to $H_eE$ is an isomorphism $H_eE\cong T_{\pi(e)}M$, which is what makes the horizontal lift of a tangent vector well-defined and unique.

Examples

1. Trivial bundle horizontals. For $E=M\times F$, the choice $H_{(x,f)}E:=T_xM\oplus\{0\}$ is a horizontal subbundle complementary to $V_{(x,f)}E=\{0\}\oplus T_fF$.
2. Horizontal subbundle on a principal bundle. If $P\to M$ carries a principal connection, the horizontal subbundle is the kernel of the connection 1-form inside $TP$.
3. Horizontal directions in a vector bundle. Given a linear connection on a vector bundle $E\to M$, there is a canonical splitting of $TE$ into “vertical directions” (changing the vector in the fiber) and “horizontal directions” (moving in the base while keeping the vector parallel).