Vertical vector field of a surface unit tangent bundle
= Vertical vector field of a surface unit tangent bundle
{title2=$V=\partial_\vartheta$}
On an oriented <Riemannian surface>, the vertical vector field generates positive rotation in each fibre of its <unit tangent bundle>: $Vf(x,v)=\left.\partial_t f(x,\cos t\,v+\sin t\,iv)\right|_{t=0}$. Hence $V(v)=iv$ and $V(iv)=-v$. In the canonical frame, $[V,X]=H$, $[V,H]=-X$ and $[X,H]=KV$.