Sliding motions on systems with non-Euclidean state spaces: A differential-geometric perspective

The paper’s Corollary 3 states that a time-dependent vector field f̃ on X̃ descends uniquely to a vector field f on X = X̃/Γ provided the Lie derivative L_{f̃} of every Γ-basic function is invariant along Γ-orbits; the proof defines f(x,t) = q_* f̃(x̃,t), shows independence of the lift via this Lie-derivative criterion, and asserts smoothness via local coordinates . The candidate solution implements the same criterion, but makes the derivation–tangent identification and the smoothness via local sections explicit (using that q is a surjective submersion under a free, proper action), which aligns with the paper’s setup on quotients and submersions , . No logical conflicts were found; the candidate fills in technical details that the paper sketches.