 Horizontal lift of a vector field through a submersion

ID: horizontal-lift-of-a-vector-field-through-a-submersion

After choosing a Riemannian metric on the source of a submersion

F : X \to Y

H_{p} = (ker D_{p} F)^{⊥}

map isomorphically to

T_{F (p)} Y

. Therefore every vector field

w

Y

has the smooth lift

v (p) = (D_{p} F ∣_{H_{p}})^{- 1} w (F (p)) .

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