Horizontal lift of a vector field

Let $\pi:E\to M$ be a surjective submersion with an Ehresmann connection and horizontal spaces $H_eE\subset T_eE$.

Let $X$ be a vector field on $M$. Since $d\pi_e:H_eE\to T_{\pi(e)}M$ is an isomorphism for each $e$, one can lift $X$ pointwise and obtain a vector field on $E$.

Definition. The horizontal lift of $X$ is the unique vector field $X^{\mathrm h}$ on $E$ such that:

1. $X^{\mathrm h}(e)\in H_eE$ for every $e\in E$, and
2. $d\pi_e(X^{\mathrm h}(e)) = X(\pi(e))$ for every $e\in E$.

Equivalently, $X^{\mathrm h}(e)$ is the horizontal lift of the tangent vector $X_{\pi(e)}$ at $e$, for each $e$.

Examples

1. Product bundle. On $E=M\times F$ with product horizontals, the horizontal lift of $X$ is $(X,0)$: it differentiates in the base direction and does nothing in the fiber direction.
2. Horizontal lift on a principal bundle. Given a principal connection on $P\to M$, the lift $X^{\mathrm h}$ is the unique $G$-equivariant vector field on $P$ that is everywhere horizontal and $\pi$-related to $X$.
3. Lift to the tangent bundle. For an affine connection on $TM\to M$, the horizontal lift of $X$ is a vector field on $TM$ describing infinitesimal parallel translation of tangent vectors along the flow of $X$.