Horizontal lift of a vector field - Imported Knowlpedia Content

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:

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

Equivalent characterizations

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

Examples

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.
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 $X$ .
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 $X$ .