Vertical tangent space

Let $\pi:E\to M$ be a fibered manifold and let $e\in E$ with $x=\pi(e)$ . The vertical tangent space at $e$ is

V_eE \;:=\; \ker(d\pi_e)\subset T_eE.

Equivalently, viewing $d\pi$ as a smooth bundle map $d\pi:TE\to TM$ between the tangent bundles, $V_eE$ is the kernel of $d\pi$ on the fiber over $e$ .

By the submersion theorem, the fiber $E_x=\pi^{-1}(x)$ is an embedded submanifold of $E$ , and

T_e(E_x)=V_eE.

In particular, $\dim(V_eE)=\dim(E)-\dim(M)$ , so the vertical tangent space has constant dimension along each connected component of $E$ . As $e$ varies, the spaces $V_eE$ assemble into the vertical subbundle $VE\subset TE$ .

Examples

Product projection: for $\pi=\mathrm{pr}_1:M\times F\to M$ , one has $V_{(x,f)}(M\times F)\cong \{0\}\times T_fF$ .
Tangent bundle: for $\tau:TM\to M$ and $v\in T_xM$ , the vertical space $V_v(TM)=\ker(d\tau_v)$ is canonically identified with $T_xM$ (it is the tangent space to the fiber $T_xM$ at $v$ ).
Principal bundle: if $\pi:P\to M$ is a principal G-bundle with structure group $G$ and Lie algebra $\mathfrak g$ , then $V_pP$ is the tangent space to the orbit $p\cdot G$ , and the fundamental vector field construction gives a linear isomorphism $\mathfrak g\cong V_pP$ .