Vertical vector field

A vector field on the total space of a fibered manifold that is tangent to every fiber.

Vertical vector field

Let $\pi:E\to M$ be a fibered manifold . A vector field $X$ on $E$ is vertical if for every $e\in E$ ,

X_e\in V_eE\quad\text{equivalently}\quad d\pi_e(X_e)=0,

where $V_eE$ is the vertical tangent space at $e$ .

Equivalently, $X$ is a smooth section of the vertical subbundle $VE\subset TE$ . Any integral curve of a vertical vector field lies entirely inside a single fiber $E_x$ . Consequently, wherever the local flow of $X$ is defined, it yields fiberwise local diffeomorphisms of $E$ covering $\mathrm{id}_M$ ; in particular, each time- $t$ map is a fiber-preserving map over $\mathrm{id}_M$ .

Examples

Product projection: on $M\times F\to M$ , any field of the form $X_{(x,f)}=(0,Y_{(x,f)})$ (no component along $M$ ) is vertical; for instance $X(x,f)=(0,Y_f)$ for a fixed vector field $Y$ on $F$ .
Principal bundles: on a principal G-bundle $P\to M$ , each $\xi$ in the Lie algebra $\mathfrak g$ defines a vertical fundamental vector field $\xi_P$ generating the right $G$ -action.
Vector bundles: on a vector bundle $E\to M$ , the Euler (radial) vector field that differentiates fiberwise scalar multiplication $t\cdot e$ is vertical.