Horizontal lift of a vector field through a submersion
= Horizontal lift of a vector field through a submersion
After choosing a <Riemannian metric> on the source of a submersion $F:X\to Y$, the orthogonal complements $H_p=(\ker D_pF)^\perp$ map isomorphically to $T_{F(p)}Y$. Therefore every vector field $w$ on $Y$ has the smooth lift
$$
v(p)=(D_pF|_{H_p})^{-1}w(F(p)).
$$